I went to civilwardata.com and searched, but did not find a T.T. among the 164 soldiers listed. There are two with a first name starting with T. and have no middle initial or mid name.
Enl 3-1-1862 at Appomattox County, VA
mustered into H Co. VA 18th Infantry
killed 9-17-62 Antietam, MD
enl 5-7-61 at Appomattox County, VA
WIA 6-30-62 Frayser's Farm
POW 7-3-63 Gettysburg, VA
absent sick 8-15-64 (intermittant fever)
POW 4-6-65 Sailor's Creek, VA
paroled 6-5-65 Point Lookout, MD
If neither of these fit, are you positive of TT? There are a number of JT and TJ, a J being hard to distinguish from a T in a florid hand.