Exhibit 99.1

TEL OFFSHORE TRUST

THE BANK OF NEW YORK MELLON TRUST COMPANY, N.A., CORPORATE TRUSTEE

919 CONGRESS AVENUE / (800) 852-1422 / AUSTIN, TEXAS 78701

TEL OFFSHORE TRUST ANNOUNCES THERE WILL BE NO SECOND QUARTER 2013 DISTRIBUTION

AUSTIN, TEXAS JUNE 28, 2013—TEL OFFSHORE TRUST announced that there will be no trust distribution for the second quarter of 2013 for unitholders of record on June 28, 2013. The Trust has not been able to make a distribution to unitholders for eighteen consecutive quarters, or since January 9, 2009. Except as disclosed below, the financial and operating information included herein for the Trust’s second quarter of 2013 reflects financial and operating information with respect to the royalty properties for the months of February, March and April 2013.

Gross proceeds for the royalty properties exceeded development and production costs for the months of February, March and April 2013 by $1,993,763, or $498,441, net to the entire overriding royalty interest, and $398,753 attributable to the Trust.  The foregoing also includes and reflects revenue being reported for the first time from a new well at Eugene Island 339 for the months of December 2012 and January 2013.  The excess proceeds, net to the entire overriding royalty interest, were applied to reduce the accumulated excess cost carryforward, which represents the amount by which the aggregate development and production costs for the royalty properties since November 2008 have exceeded the related proceeds of the production.

Gas revenues recorded by Chevron, as a Working Interest Owner, on its royalty properties decreased approximately 96% to $80,058 in the second quarter of 2013 from $1,922,266 in the first quarter of 2013, as the first quarter benefitted from the audit adjustment for Eugene Island 339 which resulted in an increase of $1,884,306 in gas revenues.  Although the second quarter benefitted from production from the new wells in Eugene Island 339 of 6,091 mcf, natural gas volumes during the second quarter of 2013 decreased approximately 88% to 21,159 Mcf from 172,227 Mcf during the first quarter of 2013.  This decrease also results from the first quarter benefit of an audit adjustment for Eugene Island 339 resulting in an increase in production of 162,885 Mcf recorded for the first quarter of 2013.  The average price received for natural gas production decreased approximately 2% to $3.78 per Mcf in the second quarter of 2013 as compared to $3.85 (excluding the effects of the first quarter audit adjustment) per Mcf in the first quarter of 2013.

Crude oil and condensate revenues recorded by Chevron, as a Working Interest Owner, on its royalty properties decreased approximately 5% to $3,654,793 in the second quarter of 2013 from $3,852,733 in the first quarter of 2013, which included the benefit of the audit adjustment for Eugene Island 339. Although the second quarter benefitted from revenue of $1,029,665 from two new wells in Eugene Island 339 for the production period from December 2012 to April 2013, this is partially offset by the audit adjustment for Eugene Island 339 in the first quarter that resulted in an increase of $852,014 in crude oil and condensate revenues.  Crude oil and condensate volumes during the second quarter of 2013 decreased approximately 33% to 33,195 barrels, compared to 49,840 barrels reported in the first quarter of 2013. The second quarter

benefitted from production from the new wells in Eugene Island 339 of 9,427 barrels from December 2012 to April 2013.  This production from the new wells at Eugene Island 339 is offset by the audit adjustment included in the first quarter of 2013 for  Eugene Island 339 which had resulted in an increase in production of 19,958 barrels recorded for the first quarter of 2013. The average price received for crude oil and condensate production increased approximately 10.0%  to $110.10 per barrel in the second quarter of 2013 as compared to $100.42 (excluding first quarter audit adjustment) per barrel in the first quarter of 2013.

Capital expenditures decreased by $91,906 in the second quarter of 2013 to $74,103, as compared to $166,009 in the first quarter of 2013. Operating expenses increased by $336,763 in the second quarter of 2013 to $1,529,678 as compared to $1,192,915 for the first quarter of 2013.

As previously disclosed, during September 2008, the platforms, wells and infrastructure associated with the Eugene Island 339 field were completely destroyed by Hurricane Ike.  While Chevron has completed the work it deemed required to clear the remaining infrastructure and abandon those wells, platforms and infrastructure, the interruption in production and the plugging and abandonment costs, each related to the Eugene Island 339 field, were the primary contributors to the resulting excess of development and production costs over proceeds of production.  Chevron has informed the Trust that the estimate of the Trust’s net portion of the aggregate cost to plug and abandon the wells, remove and abandon platforms and infrastructure and remediate the surface subject to the overriding royalty interest on Eugene Island 339 as of January 31, 2013 remains at approximately $19.8 million, and this is the same amount as the estimated cost as of October 31, 2012.  Of this amount, approximately $19.76 million has been incurred through March 31, 2013.  While this work may be complete, additional vendor invoices for this work may be received in the future, which expenses will or could bear upon the proceeds which may be received from Eugene Island 339.

As discussed above, gross proceeds for the royalty properties for the second quarter of 2013 exceeded development and production costs thereof, with the portion of such excess attributable to the Trust’s remaining 20% royalty interest equal to approximately $398,753. As of April 30, 2013, after applying the net proceeds for the months of February, March and April 2013 (and December 2012 and January 2013 with respect to Eugene Island 339), the amount by which the aggregate development and production costs for the royalty properties since November 2008 have exceeded the related proceeds of production from the royalty properties has been reduced to approximately $6.4 million (or $5.2 million, net to the Trust’s remaining 20% royalty interest) as compared to $7.9 million (or $6.3 million, net to the Trust’s remaining 20% royalty interest) as of October 31, 2012. The Trust will not receive any distribution of net proceeds until such time as the proceeds of production exceed the accumulated excess cost carryforward. As a result, the Trust will not be receiving any net proceeds for the second quarter of 2013.

In 2011, Chevron, as the Managing General Partner of the TEL Offshore Trust Partnership (the “Partnership”), consummated the sale by the Partnership of 20% of the Partnership’s overriding royalty interest (a 25% net profits interest in certain oil and gas properties) to RNR Production, Land and Cattle Company, Inc. (“RNR Production”), as the Trust needed funds to pay for liabilities of the Trust. The Partnership retained a 20% royalty interest (representing 80% of the original 25% interest).

As previously reported, in July 2012, the Trustees provided written notice to Chevron that the Trust needed funds to pay for the liabilities of the Trust and therefore instructed Chevron, as the Managing General Partner of the Partnership, to take certain actions involving a potential sale by the Partnership of a portion of its overriding royalty interest (a 20% net profits interest in certain oil and gas properties).  The Trustees and Chevron initiated the process to conduct an auction and sale of a portion of the overriding

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royalty interest similar to that in 2011.  However, in December 2012 following the due date for any bids for the purchase of the overriding royalty interest, the Trustees elected to delay any further action with respect to the proposed sale by the Partnership of the overriding royalty interest.  Since the action to delay any proposed sale by the Partnership of the overriding royalty interest, the Trustees have continued to evaluate the alternatives available to the Trust to obtain the funds needed to pay the liabilities of the Trust.  In May 2013, the Bank of New York Mellon, N.A., as an affiliate of the Corporate Trustee, made a cash advance of $300,000 to the Trust to enable the Corporate Trustee to pay administrative expenses.  The advance is evidenced by an unsecured Demand Promissory Note that bears interest at the rate of one-half percent (0.5%) and is due and payable upon demand, or if no demand is made, on May 23, 2014  The Trust Agreement provides that in the event of any such borrowing by the Trust, the Trustees shall suspend any further distributions until the advances and loans are repaid in full.  Despite the loan from Bank of New York Mellon, N.A., the Trustees continue to evaluate the alternatives available to the Trust to obtain funds necessary to pay the liabilities of the Trust which may include initiating a sale by the Partnership of its remaining interest in the overriding royalty interest.

This press release contains forward-looking statements.  Although the Managing General Partner of the TEL Offshore Trust Partnership has advised the Trust that the Managing General Partner believes that the expectations contained in this press release are reasonable, no assurances can be given that such expectations will prove to be correct. The Working Interest Owners alone control historical operating data, and handle receipt and payment of funds relating to the royalty properties and payments to the Trust for the related royalty. The Trustees of the Trust cannot assure that errors or adjustments by the Working Interest Owners, whether historical or future, will not affect future royalty income and distributions by the Trust.  Other important factors that could cause these statements to differ materially include delays and costs in connection with repairs or replacements of hurricane-damaged facilities and pipelines, including third-party transportation systems, the actual results of drilling operations, risks inherent in drilling and production of oil and gas properties, the Trust’s ability to pay its liabilities, and other factors described in the Trust’s Form 10-K for the year ended December 31, 2012 under “Item 1A. Risk Factors,” as well as other risks identified from time to time in its reports on Form 10-Q and Form 8-K as filed with the Securities and Exchange Commission.  Statements made in this press release are qualified by the cautionary statements made in these risk factors. The Trust does not intend, and assumes no obligations, to update any of the statements included in this press release.

The Bank of New York Mellon Trust Company, N.A.

AS CORPORATE TRUSTEE

CONTACT:  Mike Ulrich

(800) 852-1422

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