Now, although admittedly circumstantial, perhaps here is a trail to follow regarding how your Pvt Thomas J. Cash, 56th Georgia, arrived at Mobile after surviving Vicksburg.
In looking up Pvt Cash's service records, as you'd noted earlier, his name appears on a "List of sick and wounded Confederate Prisoners delivered to the Confederate authorities at Mobile, Ala.” but, in his case the archivist fortunately left a small clue in the footnote of that particular record. That notation reads: “Indorsement shows: ‘Rec’d 347 of the above named men. – R.C. Forsyth, Lt.-Col., P.A.C.S’”
This identical record (and footnote) appears in the file of a man I’ve researched, Pvt. C. Stevens, (37th Alabama). His records show that Pvt. Stevens was another of those 347 men arriving together at Mobile, BUT Stevens also has another record in his file that points to how he got to Mobile … that record indicates that he is named on a “List of Surgeons, Ass’t Surgeons, Attendants, and sick Confederate Prisoners of war captured at the capitulation of Vicksburg, Miss., July 4, 1863; now on board the U.S. Steamer H. Chouteau en route for Mobile, Ala., via New Orleans, La. List dated Hdqrs. Vicksburg, Grant’s Army, —.”
So, it seems to me to be a very strong likelihood that Privates Cash and Stevens shared the trip aboard the H. Chouteu alongside 345 other sick and wounded men and their caregivers.
-Chip Culpepper in Little Rock