There are two other songs regarding Big Jim. These two were done durying his '62 campaign. I have both of these on 45 rpm record.
The Ballad of Big Jim Folsom(to the tune of 'Big John')
Big Jim Folsom's Gonna Be That Man Again

in the '46 campaign, there is a 78 rpm record recorded. (though it may have been during the '54 campaign)
I saw a photo of it in a website that the writer (Bob Ingram?) donated to the Alabama Archives(please correct me if i'm wrong on these facts).