Jack, since Confederate volunteers in Arkansas tended to join companies organized near where they lived, the most likely candidate would be the James D. Barnett who served in Co. C, 3rd Arkansas Infantry; however, this man died during the war, not after.

He enlisted at Monticello, Arkansas, on June 10, 1861, and was mustered into service with his company at Lynchburg, Virginia, on June 20, 1861, for three years or the war. He died of disease in a hospital at Fredericksburg, Virginia, on March 17, 1862. I've seen his age [in 1861] listed as both 22 and 27.