That Liza Jane, po gal, really got around! She seems to have completely overshadowed our mystery man, Limber Jim, who might have been a Yankee prisoner at Andersonville (apparently he was on tv a few years ago)- a historical figure - or perhaps an outlaw in the Indian Territory in the 1870's, whose real name was J. Samuel Merrick, or some guy who was shot in Cheyenne, Wyoming, along about the same time, or one of several race horses in Missouri, or a locomotive in West Virginia, or I believe even a contemporary felon (all of the above floating around in another long GOOGLE search). Oh, I forgot the guy who wandered off into a swamp in Indiana, whose was called Limber Jim, and of course when Limber became lost, we get the "Limberlost"...and he never returned. But back to the song. Here are a few more "echoes".

From White's AMERICAN NEGRO FOLK-SONGS, pp. 242-43:

Over de hills and a great way off
De jaybird died with de hookin' cough,
An a way down in a big muddy pon'
Oh dis jaybird died wid his breeches on.

Or

'Way down yonder, and a long way off,
Jaybird died wid der whoopin' cough.

And, from Thomas Talley, pp. 153 and 212:

De jaybird build on a swingin' lim',
De sparrow in de gyardin;
Dat ole gray goose in de panel o' de fence,
An' de gander on de t'other side o' Jordan.

and,

A-way down yoner at de ris-in sun,
Throwin' san' on me!
Dat Jay-bird talkin wid a forked tongue;
Throwin' san' on me!
He's been down dar whar de Bad Men's dwell,
Throwin' san' on me!
"Ole Fri-day Devil" far' you well!
Throwin' san' on me!

{Some folks believed that the Jaybird went down to Hell every Friday to get his hair done. From the sound of about five of them outside my window this morning I could believe that they just go back!}

From the Frank Brown collection of North Carolina Folklore, Vol. III, p. 202,

The jaybird sat on the redbird's nest.
The redbird sat and mourned.
The blind man sat and shopped his shoe,
And the boatman blowed his horn.

And,

Redbird sitting in jaybird's nest,
Jaybird sitting in de morn.
Oh, look at the blind man sewing up the shoe
And the dead man just coming to.

And back on the previous page, 201,

Jay bird died with the whoopingcough,
Black bird dies with the colic;
'Long came a toad-frog with his tail bobbed off
And that broke up the frolic.

And I like,

Way down yonder, a long way off,
A jay bird died with the whooping cough.
Stiff shirt collar, three rows of stitches,
Square-toed boots and short-legged breeches.

So you can see how that broken-hearted redbird might have gotten together with that bob-tailed bull frog...

There is a series of articles in the JOURNAL OF AMERICAN FOLKLORE by E.C. Perrow, #25(1912), pp.137-55; #26(1913), pp.123-73; and #28(1915), pp.129-190, entitled "Songs and Rymes from the South" that is worth looking at. And once you get in there you'll be lost for the rest of the day discovering stuff that no one has looked at in nearly a hundred years. From Volume XXVI, p.125, from Mississippi:

I hitched my horse to the poplar trough,
The poplar trough, the poplar trough, the poplar trough,
And dar he cotched de whoopin'-cough,
De whoopin'-cough, de whoopin'-cough, de whoopin'-cough.

I hitched my horse to the swingin' lim, etc.
And dar he cut de pidgin-wing, etc.

And finally, from STEP IT DOWN: GAMES, PLAYS, SONGS, AND STORIES FROM THE AFRO-AMERICAN HERITAGE by Bessie Jones and Bess Lomax Hawes, pp. 100-103:

Way down yonder,
Sometimes,
Below the log,
Wild geese are holl'ring.
Sometimes,
Ganders trot,
Sometimes,
Bullfrog marry,
Sometimes,
His mother-in-law,
Sometimes,....

And,

Way down yonder,
Soup, soup,
Below the moon,
Soup, soup,
I got a letter,
Soup, soup,
From Alma Stone....

And, on pp. 119-120,

Way down yonder where I come from,
Oh, happy land,
Girls love boys like a hog loves corn,
Oh, happy land,
Way down yonder in the old cornfield,
Oh, happy land,
Black snake popped me on my heel,
Oh, happy land,

In each of these songs, both Bessie Jones and Alan Lomax made "adaptations" and added new words and "material". A more traditional version might be found on pp. 134-35:

Way down yonder,
Sandy ree,
Where I come from,
Sandy ree,
Girls love boys,
Sandy ree,
Like a hog loves corn,
Sandy ree.

A version of "Sangaree".

When you look at all this stuff, it is not hard to begin to imagine how "Limber Jim" aka "Buckeye Jim" emerged. Nor is it hard to imagine singing it a number of different ways, fast or slow, call and response, etc.