According to their records, it appears that the St. Maurice sailed with them from Vicksburg on or about 30 July 1863 bound for Mobile AL via New Orleans LA arriving at Fort Morgan (mouth of Mobile Bay, Mobile, AL) on 5 Aug 1863. Whether the St. Maurice made multiple trips is unclear, but at least one transport ship (the H. Chouteau ) is known to have made at least two round trips during this time period (see below).
SOURCE: Combined Service Records of these men of the 37th Ala Inf:
S. Jasper Bedgood, David Clark, James M. Farris, Benjamin Norsworthy, Daniel Rogers and Levi N. Owen.
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Service records of other men of this same regiment show the U.S. steamer Suffolk sailed on or about 13 July 1863 from Vicksburg and delivered 181 sick and wounded men and 21 officers to Fort Morgan on 22 July via New Orleans.
Additionally, the steamer H. Chouteau appears to have sailed on/about 15 July with 197 sick aboard to Mobile via New Orleans, and this ship made at least one more trip with another 347 aboard, following the same route.