George, my primary research subject, the 37th Alabama Infantry had a number of men transported to area hospitals in and around Mobile via New Orleans following Vicksburg. I've identified four steamers involved in their transport (one of which appears may have made a couple of trips to do so):

The U.S. steamer Suffolk sailed from Vicksburg on or about 13 July 1863 and arrived at Fort Morgan (at the mouth of Mobile Bay) with 181 enlisted men and 21 officers -- all sick or wounded POWs from Vicksburg via New Orleans.

The H. Chouteau sailed on or about 15 July 1863 with 197 sick/wounded POWs aboard en route to Mobile. This same vessel arrived at Mobile on 27 July 1863 with a total of 347 sick/wounded men aboard bound out of New Orleans.

The S.S. Crescent left Vicksburg with 611 aboard on or about 16 July 1863 and arrived at Mobile Harbor on 4 August 1863.

The St. Maurice departed on or about 30 July from Vicksburg and arrived at Fort Morgan on 5 August 1863 (though, no total number of men transported is recorded that I've yet located).

Perhaps one of the transportation experts on this forum can add more detail/insight.

Cordially, Chip (also in Little Rock)