The only version of this tune that I ever heard was one that my dad, grandpa, and uncles would sing - one verse and one verse only (or maybe it's only half a verse). The verse may have been something that was either made up by my paternal side of the family or someone that they knew who then taught it to them. It seems that it was meant to be a parody of the more serious versions that I have seen posted above, though I have NO idea what it was supposed to mean, other than just being silly. If there were more to the parody, I'd love someone to complete it for me. Singing it always made me laugh, or at least brought a smile. In a 6/8 feel (like "Irish Washerwoman") it went something like this:

The (F) elephant went down the track,
Goodbye, my lover, good (C7)bye,
(just a-) Pickin' his teeth with a carpet tack,
Goodbye, my lover, good (F)bye.

Chorus:
(F)Bye, Oh, my baby, there's
(C7)one more river to (F)cross,
(F)Bye, Oh, my baby, there's
(C7)one more river to (F)cross.