The surviving records of Confederate hospitals are few and far between. The first location you should check is the soldier's Compiled Military Service Record (CMSR). The "cards" (with the exception of original records in the record) refers to an another original record that his name appears on. So, therefore, if Private Johnny Reb's name appears on a "card" then look down at the bottom it should give some numbers (usually Chapter, volume, page number) that point to where his name was taken from.
Some records have survived outside the National Archives. Many of the Army of Tennessee's medical administration records that did survive were thanks to the efforts of Surgeon Samuel H. Stout. A very good book on the subject is "Confederate Hospitals on the Move : Samuel H. Stout and the Army of Tennessee" by Glenna R. Schroeder-Lein, published 1994. One of Stout's daughters sold portions of the records to various collectors and archives over the years and they are pretty dispersed. Below are several known locations of records noted by the National Archives for Chattanooga Hospitals, however, there are a lot more Chattanooga Hospitals and this does not include the Catoosa Spring hospital. The book referenced above gives several locations of his records in the bibliography.
Foard Hospital (Chattanooga): A register of patients, November 2, 1861-March 15, 1863 (one volume), is in the Texas Archives at the UT Library (S.H. Stout Collection)
Newsom Hospital (Chattanooga): A register of patients, July 26, 1862-July 14, 1863 (four volumes), The register for December 1862-January 8, 1863, includes a list of sick in wounded Federal prisoners received.
Smallpox Hospital (Chattanooga): Record Group 109 (National Archives), Orders and circular received, May 22-August 15, 1863 (Chapter VI, Volume 274, page 3-18).
I hope this helps.
Gerald D. Hodge, Jr.
War Between the State Historian
Historian: 39th Georgia Volunteer Infantry Regiment