I agree that regimental histories available on-line do not mention Perryville, however, in addition to the CW Research Data Base, Alton, Ill POW records show at least one 37th Miss. POW captured at Perryville.
Alton Prison [http://www.altonweb.com/history/civilwar/confed/j.html]
Rank, Company, State
Date and Place
Date of Death
Cause of Death
Place of Burial
Jackson, W W
From the CW Research Database
Warren W. Jackson
Residence was not listed;
Enlisted as a Private (date unknown).
He was listed as:
* POW 10/11/1862 Perryville, KY
He also had service in:
"H" Co. MS 37th Infantry (Died of variola)
Buried: Confederate Cemetery, North Alton, IL
Personnel from the 37th MS Infantry
who were wounded at Perryville, KY
4 Rows Found i.e., four deaths are listed [as I previously noted] with name, Co., etc.