Note: The next surviving company muster roll is for the period August 31 to December 31, 1862, dated January 20, 1863. As he is not mentioned, he was dropped from the rolls sometime in between, either for discharge, desertion, transfer, death, capture, etc.
M323: Compiled Service Records of Confederate Soldiers Who Served in Organizations from the State of Texas
There is a J. W. Conner, Pvt., Company A, 2nd Texas State Troops Cavalry
and two John Conner's one a Pvt. in Co. H, 24th Texas Cavalry & Pvt., Co. G, 9th Texas Infantry
I have not checked their records.