Talley. Negro folk rhymes

Title: Negro folk rhymes, wise and otherwise
Author: Thomas W. Talley
Year: 1922

Online at Project Gutenberg and at Hathi Trust.

There is so much wonderful material in this book, along with a detailed explanatory essay. You can read about Thomas Talley's ground-breaking work at Wikipedia. This was the first collection of folksongs published by an African-American researcher!

I found lots of rhymes that could qualify as chain tales because of numbers, weekdays, other sequences, etc. -- and even a version of cumulative "Bought Me a Rooster" (where the cat goes "fiddle-toe! fiddle-toe!"):

Jaybird Died with the Whooping Cough
Gray and Black Horses
Peep Squirrel
Did You Feed My Cow?
A Budget
Satan
Bedbug
How to Keep or Kill the Devil
Master Killed a Big Bull
Bought Me a Wife
When I was a Roustabout
Animal Attire
Little Boy Who Couldn't Count Seven
The End of Ten Little Negroes
Fishing Simon
Getting Ten Negro Boys Together
Little Sleeping Negroes
Frog Went A-Courtin'
Destinies of Good and Bad Children 
Animal Persecutors
A Man of Words
Jack and Dinah Want Freedom

(first edition, 1922)

Historical memorial on the Fisk University campus:

(photo by Dave at Flickr)