Governor George W Johnson served as a volunteer to General Breckenridge and Colonel Robert P. Trabue. Johnson was governor of Kentucky. Known as private with Company E of the 4th Kentucky Infantry Regiment.
Harrison, Lowell H. “George W. Johnson and Richard Hawes: The Governors of Confederate Kentucky.” The Register of the Kentucky Historical Society, vol. 79, no. 1, 1981, pp. 3–39. JSTOR, www.jstor.org/stable/23379523. Accessed 2 Feb. 2021.
Governor Isham Harris of Tennessee had joined the staff of General Albert Sidney Johnston.
Tennessee: A History, 1673-1932
(New York: American Historical Society, Inc., 1933), pp. 508
This question is worded in what could be considered a vague manner. Every governor is the other governor in someone else's point of view. Gov. Isham Green Harris of Tennessee was an aide to General Albert Sidney Johnston. Gov. George Washington Johnson of Kentucky was also an aide on the staff of a Confederate general. One source says Breckinridge , another says Buckner. Lew Wallace wasn't governor of New Mexico yet and he was acting like a major general on the Shiloh Battlefield. Louis Harvey of Wisconsin technically was not on the battlefield but he was at Shiloh when he drowned. But I'm guessing your referring to GOV. GEORGE WASHINGTON JOHNSON as the "other" governor and he was acting in the capacity of STAFF MEMBER (AIDE) TO GEN.JOHN C. BRECKINRIDGE.They were cousins you know?
Since we had an earlier Trivia question on Kentucky Governor George W. Johnson (1811- 1862), the "other governor" is Tennessee Governor Isham Harris (1818-1897), who served as a Staff Officer to General Albert Sidney Johnston.
Isham Harris, Governor of Tennessee, served on Johnston's staff and was with him when he died.
George Johnson, Confederate Governor of Kentucky, was mortally wounded while fighting with a Kentucky regiment at Shiloh.
Answer: Isham G. Harris, Governor of Tennessee serving as a volunteer aide to Gen. Albert Sidney Johnston. He was with Johnston when he was wounded until he died. Harris is the one who pinpointed the place of Johnston‘s expiration post-war.
Source: Isham G. Harris of Tennessee, by Sam D. Ellliott
Edit - George Johnson of Kentucky is also an acceptable answer. And since the question didn't rule out people who became governor of a state subsequent to the Battle of Shiloh, Henry Allen of Louisiana is also acceptable.