Mr. Waller, the following is taken from the Civil War Soldiers Index, Family Search.org (a service provided by The Church of Jesus Christ). It is the only entry I found which referenced Company H and is listed under "Conferate Troops" rather than just under Mississippi.

A.T. Roberts
United States, Civil War Soldiers Index
other: 1861-1865 Confederate Troops

name: A.T. Roberts
event: Military Service
rank in: Private
rank out: Private
side: Confederate
state (or origin): Confederate Troops
military unit: 4th Regiment, Confederate Infantry
military unit note: (1 Regiment Alabama, Tennessee, and Mississippi Inf.)
company: H
general note: See also 42 Tenn. Inf.
nara publication title: Index to Compiled Service Records of Confederate Soldiers Who Served in Organizations Raised Directly by the Confederate Government and of Confederate...Officers and Nonregimental Enlisted Men.
nara publication number: M818
nara roll number: 20