The Wm. T. Bryant in Company E of the VA 9th Infantry died 8-3-62 at Huegenot Springs, VA, so he is not your man.

One who did survive the service: William T. Bryant, VA 32nd Infantry, Co. F. Enlisted as a private on 1-14-64, was discharged for disability on 4-13-64.

One who might have survived; W.T.Bryant, VA 3rd Battalion Reserves Infantry, Co. H. No other data.

Then, I found records for nine William Bryant, with no middle initial.