Here's another military Thorn, a Private Elias Thorne, killed during the Overland Campaign in 1864 on the Union side. If he was killed on May 19th, then the action was at Harris's Farm, not Alsop's - which would have made his regiment the 8th New York Heavy Artillery, which took the by-1864-typically massive casualties of a raw, green Union regiment in a heads-up open-field fight with Confederate veterans.

Update: yep, Eighth New York Heavy Artillery. His cousin Alonzo C. Taylor was in the same unit, and apparently was wounded in the same action.