William C. Grizzle's service records indicate he died in the hospital in Jackson (Hinds County), MS.
He is noted as Grizell* Wm C, 4th regt, co F,[died] May 20, 1863 in this post-war newspaper article provided by Jeff Giambrone.
http://history-sites.com/cgi-bin/bbs53x/mscwmb/arch_config.pl?read=10807
A "possible" burial site may be the Greenwood Cemetery -