I find 4 William Loudermilk's from N.C.
Corpl. William A. age 21, residence Cherokee Co. N.C. 2nd NC Cav. injuried in Railroad accident and discharged 1/20/63.

Pvt. William D. age 22, residence Caldwell Co. N.C, 26th NC Inf. Died of wounds 6/12/64.

Pvt. William M. Age 28, residence Cherokee Co. N.C. 39th NC Inf. Deserted and dropped from rolls 2/28/1863.

Pvt. William W. Age 45, residence Cherokee Co. N.C. 25th NC Inf. Discharged for over age 8/15/1862.
Maybe with other information you can place him.
Mike