I checked NARA M347. "Unfiled Papers and Slips Belonging in Confederate Compiled Service Records" and found a few more E. Mays, but none that appear to be your man;

E.F. May, Pvt., D/3Va Bn, hosp Richmond 18 Dec 1862

E Ferdinand May, paroled Lynchburg, 13 Apr 1865, clerk for Maj. R.C. Saunders, controlling QM of Tax in Kind for State of Va.

E.P. (or G.P) May, pvt., Va. conscript, hosp, chronic diarrhea, Farmville, 3 Aug to 1 Sep 63 when released to duty

E.H/Edward H. May, gunsmith and carriage maker, Reserve Ord Dept, ANVa, enl. 9 Apr 64, Orange C.H.; present 31 Oct 64.

E.H./E.M. May, Rockingham Co., ? Va Cav., age 2o, took oath Martinsburg, Va., 10 Nov 63.

E.W. May, C/Invalid Corps, guard CS millitary prisons, Richmond, 17, 31 Jan 65.