Re 1862 Fort Pillow:
The Confederate Fort Pillow hospital in 1861/1862 was five miles down the Mississippi River at Randolph. There is a badly overgrown 19th Century cemetery on top of a hill overlooking the boat landing area there that was likely the place of burial for those who died at this hospital. But it would take a major bush whacking expedition to find any markers other than the civilian gravestones that I saw scattered about the hill top. Several of these dated back to the 1870's. Fort Pillow and this surrounding area was abandoned by the Confederates on or about June 1, 1862. The 12th Louisiana was marched overland from Camp Green through Covington to Randolph where they boarded steamers for the short trip down river to Memphis. From Memphis, they were moved into northern Mississippi.
Two members of the regiment fell out sick at Covington on the march to Randolph and died there. They are buried in the R. H. Munford Cemetery in Covington. They have barely legible, basic grave markers put there probably after the war by a local ladies memorial association.
Green's Chapel is located on the main east-west road into Fort Pillow surrounded by a modern Tennessee State Penitentiary facility. I did not find any obvious signs of a burial space for indigents or unmarked war time military graves. However, the 12th Louisiana Infantry moved away from Fort Pillow in early May 1862 to a camp ground about 7 to 8 miles east of Fort Pillow identified in the military records as "Camp Green." Since no senior officers in the regiment, or other famous people associated with the regiment, were surnamed "Green", I have surmised that the name had local roots - Green's plantation or farm, Green's Creek, etc. Green's Chapel Cemetery is about 7 to 8 miles east of Fort Pillow and an obvious place to look. However, I have tried, without success so far, to link the Green's Chapel Cemetery with Camp Green.
From your description of the after math of the 1864 assault on Fort Pillow, the Confederates withdrew eastward (got my directions right this time!!) towards Jackson, Tennessee. Green's Chapel should have been on their route.
I signed up with the local Lauderdale County genealogy list online and posted my query, but never made a connection with anyone who knew much about the mid 19th Century history of that area.
But, hope springs eternal!