I found two possible matches under burials at Soldier's Rest in Vicksburg...

T.M. Hill Company B 36th Miss. Inf. died: 6/30/1863
Jacob Hill (Company B? 36th Miss. INF?) Died 12/31/1862 grave number 413
Source: http://www.nps.gov/archive/vick/city_cem/chill_h.htm

If this is a possible match it may help you to know that...

"... the 36th Infantry Regiment was assembled during the spring of 1862 and mustered into Confederate service at Corinth, Mississippi. Its members were recruited in Scott, Copiah, and Hinds counties"

At least this cuts down to three counties you need to look through to see if this holds water. The list at Soldier's rest is not a complete list so take this with a grain of salt.

May I ask if your ancestor ever settled in the western part of Louisiana. Consider this info I pulled off familysearch.org...

William T. HILL
Birth Year 1850
Birthplace MS
Age 30
Occupation Farmer
Marital Status M
Race W
Head of Household William T. HILL
Relation Self
Father's Birthplace TN
Mother's Birthplace GA

Source: Census
Place District 43, 5th Ward, Vernon, Louisiana

Spouse: Jane HILL
Daughter: Alpha HILL
Daughter: Martha HILL
Daughter: Beda HILL
Other: Alice Nolan

This is just a stab in the dark I admit but maybe the names ring a bell? Any info you can give us will help. spouses name, county of origin etc.