Capt. Fletcher's Company was Co I, of Col John Griffith's 11th/17th Arkansas Mounted Infantry. See if you can get in touch with Kenneth Byrd or Anthony Rushing, as they probably know more about the regiment than anyone.
Was you man from Amite Co? I do not see him in the 1860 Census? Or did he join in Amite County when Col Griffith came thru in the 1864 time frame?