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Non-violent felons can in some cases use long guns for hunting after a certain amount of time has passed after they serve their term. (I believe in TX it’s 5 yrs after the completion of all the term, parole probation etc.) They must apply to the state for the exception. I am unsure if they can own the guns or just possess them for hunting purposes.

Federal law prohibits them from owning or possessing, regardless of state law.

See NRA Federal Gun Law summary

The following classes of people are ineligible to possess, receive, ship, or transport firearms or ammunition:

* Those convicted of crimes punishable by imprisonment for over one year, except state misdemeanors punishable by two years or less.
* Fugitives from justice.
* Unlawful users of certain depressant, narcotic, or stimulant drugs.
* Those adjudicated as mental defectives or incompetents or those committed to any mental institution.
* Illegal aliens.
* Citizens who have renounced their citizenship.
* Those persons dishonorably discharged from the Armed Forces.
* Persons less than 18 years of age for the purchase of a shotgun or rifle.
* Persons less than 21 years of age for the purchase of a firearm that is other than a shotgun or rifle.
* Persons subject to a court order that restrains such persons from harassing, stalking, or threatening an intimate partner.
* Persons convicted in any court of a misdemeanor crime of domestic violence.

Persons under indictment for a crime punishable by imprisonment for more than one year are ineligible to receive, transport, or ship any firearm or ammunition. Under limited conditions, relief from disability may be obtained from the U.S. Secretary of the Treasury, or through a pardon, expungement, restoration of rights, or setting aside of a conviction.

Note: "punishable by", not "punished by". Actual sentence doesn't matter, only maximum allowable.