This is not from the file of a member of the 36th Cav. It is from the service file of Capt. Alsdorf Faulker of the 15th Tex. Cav. Co. G. He was in captured near Nachitoches, La. on March 31, 1864. His file contains info about where he was held and where he was exchanged.

Pg 12 Appears on a Roll Of Prisoners of War received at New Orleans for the 5 days ending April 20,
1864. Captured near Nachitoches, La. on March 31, 1864.
Pg.13 Appears on a register of Prisoner of War at New Orleans, La – Exchanged July 22, 1864
Pg 15 Appears on a Roll Of Prisoners of War transferred from New Orleans, La and exchanged at
Red River Landing July 22, 1864. Roll dated Office Commisary of Prisoners, Dept. of the Gulf, N. Orleans, La. July 31, 1864

Included in this service file are two requests Capt. Faulkner made to Col. Kilbourne, Commander of the Rampart Prison where he was detained. Col. Faulkner's handwritting is somewhat difficult to read, but it reads clear enough once you get familiar with his scrawling handwritting.
I find these requests very interesting.

Pg. 51 On June 3, 1864 from U.S. Prison Rampart St., he requests of Col. Kilbourne an 8 hour parole to attend the funeral of a child of a personal friend. The friend was absent as he was taking his brother's remains to New York for burial.

Pg. 49 -On June 10, 1864 from Rampart Prison he requests of Col. Kilbourne a 3 hour parole to vist a relative and member of his company who is very sick in the St. Louis Hospital.

I hope this might give you some new clues to follow. Good luck.
Carol Webb