Rimfire,
As listed in my SRS books - Serial #s 29779, 29790, 29733 were all issued to Company K, 6th Illinois Volunteer Cavalry in 1863 (no day or month given)
and then,
Serial # 29771 was issued or re-issued to Company D, 1st Vermont Volunteer Cavalry in March, 1865.
Looking at the Serial #s as listed, I'd go with the Illinois unit.
Two Flints