There were no designated hospitals for any particular unit from the Army of Tennessee. Men would have been triaged and sent to the appropriate level of care needed and evacuated accordingly to what hospital that had bed space at the time. The further the army retrograded south, the further south the general hospitals were moved. As described in previous posts, the book on Stout describes this in detail.

Since you are a descendant of the 36th Georgia this may be of interest to you: MAJOR ANNOUNCEMENT!!!, Weekend November 21-22, 2009: I am pleased to announce a historical walking tour conducted by myself and the Chief Historian for the Chattanooga and Chickamauga National Military Park, James Ogden III, at Sherman Reservation, time and day will be refined in the coming weeks. This walking tour will focus on the fighting on the north end of Missionary Ridge on November 25, 1863 and will include the actions of the 39th Georgia Infantry and Cumming's Brigade. This event is free to the public and is presented as an anniversary program by the National Military Park. Please be prepared for inclement weather and expect to walk.

Please let me know if you have any questions.

Respectfully,

Gerald D. Hodge, Jr.
M.A. Military History - Civil War Concentration.
War Between the States Historian
Historian: 39th Georgia Volunteer Infantry Regiment