The tune is named for Jackson, Mississippi, where it was recorded in 1930 by the Mississippi Mud Steppers, a variation of the group The Mississippi Sheiks. The Mud Steppers likely consisted of Walter Vinson on guitar, with Charlie McCoy and an unknown musician.
ms_mudsteppers

The Old Jackson Stomp is actually an old piano piece by Cow Cow Davenport, "The Cow Cow Blues".
This is one of the post-ragtime "rags" that fiddlers and mandolin players developed in the early 20th century using the syncopated rhythms of the ragtime tradition.
Other rags/blues tunes in this collections are:
    "Cherry River Rag"
    "Colored Aristocracy"
    "Dora Dean"
    "L & N Rag"
    "Pig Ankle Rag"
    "Ragtime Annie"
- all in the Tunes section and
    "East Tennessee Blues"
    "Eli Green's Cakewalk"
    "Hawkin's Rag"
    "Sister Kate"
    "Stone's Rag"
    "Woody's Rag"
- in the Legacies section and
    "Horace Hanesworth"
- in the Songwriters section.
It was also recorded by The New Lost City Ramblers. I learned it from the NLCR record.
I sometimes play it as the second tune in a medley with "Colored Aristocracy" in this section.