Below is the material at the University of Texas, Austin that is listed at the attached link that pertains to the Kingston hospitals:
http://www.lib.utexas.edu/taro/utcah/00042/cah-00042.html
Papers
Box 2G394
Kingston, Georgia:
Erwin Hospital, J. Grones, surgeon
Kingston Hospital, Thomas Mattingly, surgeon
Oversized Documents: typically these are the large ledger books.
2.325/AA38 Kingston, Georgia
Erwin Hospital, J. Grones, surgeon
Kingston Hospital, Thomas Mattingly, surgeon
Box 2.325/AA39 Kingston Hospital
2.325/OD18 (M6)
2.325/OD16 (S2)
Other Stout material is located at the following libraries, archives, or museums. You would have to inquire with each institution to see what exactly they had in their collections. My best guess is that they material you are looking for is at the University of Texas, Austin.
Atlanta, Ga. Atlanta Historical Society. Samuel H. Stout Papers, collection 342f.
Atlanta, Ga. Special Collections. Robert W. Woodruff Library, Emory University. Emory University Archives. Sameul H. Stout Papers. Collection 274.
Austin, Tx. Eugene C. Barker Texas History Center. University of Texas. Samuel H. Stout Papers.
Chapel Hill, NC. Southern Historical Collection. University of North Carolina. Samuel H. Stout microfilm collection.
Cleveland, OH. Western Reserve Historical Society. Samuel H. Stout Papers.
Durham, NC. William R. Perkins Library. Duke University. Samuel H. Stout Papers.
Nashville, TN. Tennessee State Library and Archives. Samuel H. Stout Papers. Collection THS 39. (also available on microfilm). Link to a detailed breakdown of this collection can be found at http://tennessee.gov/tsla/history/manuscripts/findingaids/ths39.pdf
New Orleans. Howard-Tilton Memorial Library. Tulane University. Joseph Jones Collection.
Richmond, VA. Virginia Historical Society. Samuel H. Stout Papers.
Richmond, VA. Eleanor S. Brockenbrough Library, Museum of the Confederacy. Samuel H. Stout Papers.
Savannah, GA. Georgia Historical Society. Samuel H. Stout Papers. Collection 764.
Washington, DC. National Archives. Record Group 48. Office of Secretary of the Interior. Record Group 109. War Department Collection of Confederate Records.
I hope this helps. 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