F'r what it's worth, my mom, who was born & raised in Georgia, used to sing "Hot corn, cold corn, swing around the jimmy-john," but she told me that the song had originally been "Bring along a demi-john," and that a demijohn was an earthenware jug, usually with whiskey in it. When she was a kid they also sang "Swing around, swing around, swing around the Jimmy-John, Swing around, swing around the Jimmy-Jimmy-John" as a kind of second verse to Hot Corn. This is exactly the words used in Gandy Dancer's ball, but they use another tune.

So. Gandy Dancer's Ball is much more recent and refers to older songs including Cotton-Eyed Joe. I think that the little chorus of "Swing around the Jimmy-John" is supposed to be something people would dance to, maybe a version of Hot Corn that had been turned into a play-party song that people danced to, although I can't remember if Mom said anything about that.... I think they just threw it in to give the feeling of an old fashioned barn dance going on.