George,
As I mentioned in my second post, the seemingly contradictory evidence may be because the 34th Mississippi [which fought at Perryville but not at Corinth] was also known as the 37th.

This is supported by the following:

Dunbar Rowland’s "Military History of Mississippi"
34TH MISSISSIPPI INFANTRY
The regiment was known as the Thirty-seventh for some time.........The regiment accompanied General Bragg's army to Chattanooga in July, 1862, marched in the latter part of August through Middle Tennessee into Kentucky, with Hardee's Corps, and participated in the battle of Perryville, October 8, 1862................

Perryville OOB
Source: Perryville by Kenneth Noe (University Press of Kentucky, 2001)

Jones's Brigade (Col. Thomas M. Jones) ?
27th MS 350
30th MS ?
34th MS 300
Btty. F, 2nd AL Lt. Arty. (Capt. Lumsden) 125 4X12#

The 37th Mississippi was heavily engaged as a regiment at the battle of Corinth, as supported by the following:

Dunbar Rowland’s "Military History of Mississippi"
37TH MISSISSIPPI INFANTRY
........After Beauregard evacuated Corinth May 29, and Bragg moved the main body of the army to Chattanooga in July, the Thirty-seventh was part of the Fourth Brigade (John D. Martin's) of Little's Division (afterward Hebert's) of the army of Gen. Sterling Price, left for the protection of Northeast Mississippi..........

Official Records Series 1, Vol XVII pt 1 Numbers 106.
Return of Casualties in the Confederate forces, Major General Earl Van Dorn commanding, October 3-5.
FOURTH BRIGADE
Staff 1 1 ...........
36th Mississippi Regiment 12 71 ...........
37th Mississippi Regiment 19 62 ...........
38th Mississippi Regiment 4 31 ...........
37th Alabama Regiment 5 35 ...........