It appears family legend may be wrong. Per 12th NY Cav roster, Lt. Walker was discharged in June 1863

WALKER, HOBART M.— Age, 21 years. Enlisted, December
18, 1862, at West Seneca; mustered in as private, Co. D, December
18, 1862, to serve three years; mustered in as second
lieutenant, Co. L, March 12, 1863; discharged, June 23, 1863.
Not commissioned.

https://dmna.ny.gov/historic/reghist/civil/rosters/cavalry/12thCavCW_Roster.pdf