Jerry, the only Arkansas soldier named Mays who served in a Randolph County company was a William Mays, who served in the 38th Arkansas Infantry. He enlisted in (old) Co. I at Rice Springs, Arkansas, on August 4, 1862, and later transferred to Co. H. I don't have any information beyond his enlistment and transfer. Don't know if he survived the war or not.
Many Randolph County men enlisted in neighboring Greene County. There was a Sergt. D. D. Mays, who enlisted in Co. H, 5th Arkansas Infantry, at Gainesville, Arkansas, on September 28, 1861; age 31. He was reported absent sick in a hospital at Nashville, Tennessee, on December 31, 1861, after which there is no further record. He is presumed to have died.