Jim senior might have been Truman’s running mate, his efforts to promote populism would have played well, as well as quiet support for the first steps towards civil rights.
Truman wasn't even on the ballot in Alabama in 1948! "A "Loyalist" group did petition governor
"Big Jim" Folsom to allow Truman electors, but Senator
John Sparkman, fearing popular defeat at the hands of the Dixiecrats and a hostile state legislature, decided against placing Truman electors on the ballot."
https://en.wikipedia.org/wiki/United_States_presidential_election_in_Alabama,_1948
Folsom may have been a racial liberal by Alabama standards but he would know it would be political suicide in Alabama to be on Truman's ticket--and anyway Truman had pretty much written off the Deep South (except for Georgia, the only Deep South state where he was the official Democratic candidate.)
And of course:
"On March 3, 1948, Folsom's name was in headlines across the nation when 30-year-old Christine Johnston, a widow who had met Folsom in late 1944 while she was working as a cashier at the Tutwiler Hotel in Birmingham, filed a paternity suit against the Governor, alleging that he was the father of her 22-month-old son.
[3] Undaunted, nine days after the suit was filed Folsom appeared on the sidewalk in front of the Barbizon Modeling School in New York City, where he kissed a hundred pretty models who had voted him "The Nation's Number One Leap Year Bachelor," attracting a crowd of 2500 onlookers and causing a traffic jam. Johnston dropped the suit in June for a cash settlement from Folsom, who years later admitted to an interviewer that he was indeed the father of Johnston's child.
[4]
"On May 5, 1948, without prior publicity, Folsom married 20-year-old
Jamelle Moore, a secretary at the state Highway Department, whom he had met during his 1946 campaign and had been dating and seeing "almost daily" since then.
[5]…"
https://en.wikipedia.org/wiki/Jim_Folsom
Sure, that didn't prevent him from later being re-elected governor of Alabama--remember that's the state that almost elected Roy Moore to the Senate! But I'm quite sure that it's enough to keep him off a national ticket.