Mahlon, you're correct. Although his name was carried on the rolls of Co. B, 36th Arkansas Infantry, as William C. Murphrey, he was in fact William Carroll Murphree, born in 1839, a resident of White County, Arkansas, when the war began. He moved to Oklahoma at some point after the war, and applied for a Confederate pension there (Pension File No. 3247-09).
Before enlisting in the 36th Arkansas Infantry, he had been a member of the White County Militia (21st Regiment Arkansas Militia).