The civil war soldiers and sailors system lists 12 Thomas Conners who served in the War Between the States from Missouri (3 CS, 9 US) (no Thomas J.).
Assuming your ancestor was one of the CS and that he was captured and died in 1862 then I would bet on the Thomas Conner that was in the 3rd Missouri Infantry. The 3rd was active in the Corinth campaign in 62 and many were recruited from his part of the State (East Central Missouri)
To confirm this you'll need to get his compiled service record from the national archives.