The "Senatobia Invincibles" were originally Company I, 9th Mississippi Infantry Regiment, formed in Feb 1861 with a 12 month enlistment.
The company was reorganized in May 1862 as Company B, 42nd Infantry Regiment, and was sent to Virginia as part of the Army of Northern Virginia. When you look at the roster linked to below you will find a "T. H. Hall listed. As you probably know, "A" and "H" are often confused with each other on old documents. (and tombstones)
T. H. Hall is also listed in B Company 42nd Infantry Regiment in the Soldiers and Sailors System.
A page at the Desoto County Mississippi Genweb site about the 42nd Miss Infantry including rosters of B, C, and D Companies. http://www.msgw.org/desoto/military/42MsInfCoBCD.html
A record of the 42nd Infantry Regiment by Dunbar Roland found on the Mississippi SCV site. http://www.mississippiscv.org/MS_Units/42nd_MS_INF.htm
A portion of a website dedicated to the 42nd Miss Infantry. http://members.cox.net/rb2307/