You need to get a copy of this man's Compiled Military Service Records either from the National Archives or from the service offered by this message board.
The Battle of Perryville was fought in the fall of 1862. The Dix-Hill Cartel of July 1862 (the exchange agreement) provided that all prisoners of war were to be exchanged, or released on parole, within 10 days of their capture. While the "10 days rule" may not have been strictly adhered to, these prisoners were most likely sent to the Louisville Military Prison, and then down river to the exchange point at Vicksburg. Or they could have been released on parole near the battlefield a few days after the fighting stopped and returned to their own side. This is how the POWs taken at Corinth, Mississippi in October 1862 were handled.
The actual details should appear in the CMSR of the individual soldier. When you get the CMSR information, I would be interested in hearing back what the record actually says about their transfer and release.
Fort Delaware Society