Here are a number of men who could be your subject.
Do you have anymore personal information on him which could help in his ID?
William A. Thompson
Enlisted as a Private
"K" Co. AL 43rd Infantry
The Forty-third was organized at Mobile in May, 1862, and went
without delay directly to Chattanooga, where it was brigaded
under General Leadbetter.
Additionally, there are 21 William (nmn) Thompsons in Alabama Units, 6 W. Thompson's and 4 W. A. Thompsons
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W A. Thompson
Enlisted as a Private
"A" Co. AL Lewis' Battn Cavalry
Lewis' battalion served in central Alabama and Georgia during
the summer and fall of 1864, and until the close of the war.
It consisted of five companies under Captains Harrell, Brooks,
Morrison, Barnes and May.
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W A. Thompson
Enlisted as a Private
"C" Co. AL 4th Infantry
The Fourth Alabama regiment, commanded by Col. Egbert I. Jones, was organized at Dalton, Gal, May, 1861, and immediately proceeded to Virginia.
.....
W A. Thompson
Enlisted as a 1st Sergeant
"C" Co. AL 27th Infantry
The Twenty-seventh Alabama regiment was organized at Fort Heiman, in Tennessee, in the winter of 1861.
It was sent to Fort Henry, then to Fort Donelson, where it was
captured, though many of the command, being sick in the hospital, escaped the surrender and joined a Mississippi regiment. The captured men were exchanged in September, 1862, and were at Port Hudson during the winter.
.....
W A. Thompson
Enlisted as a Private
"C" Co. AL 29th Infantry
The Fourth Alabama battalion was organized during the fall of
1861, at Montgomery.
In February, 1862, two companies were added and the organization became known as the Twenty-ninth Alabama. It was drilled in artillery practice; remained at Pensacola until the evacuation; was between Pollard and Pensacola and at Mobile for about a year, in Cantey's brigade, which was transferred to the army of in the spring of 1864, and took part in the battle of
Resaca, May 13th, where it fought brilliantly.
Historical Data Systems, Inc.
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William A. Thompson, Private, Company K, 43rd Alabama Infantry, enlisted May 14, 1862 at Tuscaloosa, Ala. by Jno. J. Harris for 3 years or the war, discharged from hospital, November 5, 1862, "Pol. Pris.," swore to the Oath of Allegiance to the United States January 6, 1864
Note, his record notes his place of residence which I am unable to read on-line. It looks like Hudkis [sp?] County, Ala.
I suspect he deserted and turned himself in to the Yankees
M311: Compiled Service Records of Confederate Soldiers Who Served in Organizations from the State of Alabama