I would believe he would have been buried in the Thornrose Cemetery as the fellow below who died the next day. His records spell his name with one 'G', Wigington. His widow, Martha L. Wigington filed claim for his effects and accounts due.

Brown, Benager T.- 2d Lieutenant 38th Georgia, died at Staunton, Va. July 27 or 29, 1862 . Buried there in Thornrose Cemetery.

Thornrose Cemetery, Staunton, Virginia

This beautiful cemetery hosting the designs of T.J. Collins was established in 1849. At the highest elevation of Thornrose Cemetery is a large Confederate section that is marked by a monument of a soldier, impressive limestone retention walls and cast iron urns. Some 1,777 Confederate soldiers are buried in this section, the remains of many of them moved to this site following the war.
http://www.visitstaunton.com/attractions/attractions_museums.htm

William B. Booth was a private in the 26th Georgia (Co. H or K) from Ware County, Georgia, He was wounded at 2nd Manassas, and died on 12 March 1863 at the military hospital in Staunton, VA and was buried there. His Great, Great Grandson William H. Booth, Jr. reports:

I traveled to Staunton this past December (1998) to find his grave. I discovered the hospital where he died was the Virginia School for the Deaf and Blind, which was converted to a military hospital during the War. After the War, it reverted to its original function and is still the state school. According to the local historical society, soldiers brought to the hospital who died were held at the hospital for a period of time to give relatives a chance to claim the bodies. Most were never claimed because of the distances from their home states. Most of those who were unclaimed were buried in a designated area of Thornrose Cemetery in Staunton. The state of Virginia erected a large monument to the Confederate soldiers buried there, more than 1500 with more than 700 unidentified. Only three graves are marked but a directory exists. My gggrandfather was buried there 12 September 1863, six months after his death. It is an incredibly beautiful site, located on top of a hill with huge oaks that must have been there more than a century.
http://users.erols.com/brant/GeorgiaBrigade/soldiers.html