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Some very good suggestions have been made, but you should obtain a copy of his death certificate, to make sure you have full and correct death date. If, for some reason, the Tyler Pub. Lib. does not have copies, or microfilm copies, of back issues of the newspaper. Find out the name of the local newspaper for 1958, [the names change sometimes], and contact the University of TX at Austin. They have the largest collection of TX newspapers, but they do not do research. You would need the exact and correct date of death. If they find anything there will be a charge, but nominal. Also, if your father lived in San Antonio at the time, there might be something in the San Antonio newspaper, too.
If you do not live in the area, try Smith Co, TX GenWeb site, some GenWeb counties have volunteers who will do lookups, not research. If you have correct date of death, someone might visit the library and check the newspapers for you. Worth a try.
By the way, unless your father was receiving Soc Sec pension or disability, he will not be on the SSDI. That index is when Soc Sec is notified of a death. That happens when someone is receiving Soc Sec and SS must be notified to stop payments or receive a death benefit, OR, if a man dies leaving minor children, Soc Sec would be notified of his death, because those minor children will receive money until they reach 18 yrs of age.