Being listed on the census as a resident of Trenton, Ga., you might expect him to be in an Alabama unit from Jackson or Dekalb Counties, or a Tennessee unit from Hamilton County. He might also be found in the 3rd Confederate Cavalry, which included several companies from this area.
However, let me call attention to his profession, listed on the census as "railroader". If Thompson was a bona fide railroad employee of any kind, he's exempt from Confederate military service. I can check a couple of Alabama companies from Jackson County for him, but Thompson may never have enlisted at all.