I find no one by the name Edge in Jeter's Company roster. However, I do find an W. S. Edge in Ward's artillery. The initial 'S' written in cursive script can readily be read as an 'E'

George Martin

W. S. Edge, enlisted as a Private into Captain Joshua Ward's Waccamaw Light Artillery Company, South Carolina Volunteers** February 1, 1863 at George Town, last recorded present December 31, 1864

**This company subsequently became Captain Mayham Ward's Company, South Carolina Light Artillery. It was temporarily attachted to the 10th Regiment South Carolina Infantry, when mustered into service, and on August 31, 1862, it was mustered as a company of the 4th (Emanuel's) Squadron of Cavalry (subsequently 12th Battalion South Carolina Cavalry)

Ward's Company, South Carolina Artillery (Waccamaw Light Artillery)

Waccamaw Light Artillery was organized early in 1863 with men from Georgetown County. Attached to the Department of South Carolina, Georgia, and Florida, the unit served in the Charleston area until the evacuation of that city in February, 1865. It later disbanded. Captains Joshua Ward and Mayham Ward were its commanders.
http://www.itd.nps.gov/cwss/soldiers.cfm

See also: http://batsonsm.tripod.com/b/arba4.html

A Goggle Search "Waccamaw Artillery Battery Ward’s Artillery" will also bring up a number of hits