I think much of the confusion is caused by the fact that at one time there were two separate units that were both designated as the 37th Miss. Inf. Reg. -- one of these units would later be re-designated as the 34th Miss. Inf. Reg. To add to the confusion, both units, with the same designation, were at Corinth, at the same time, during the seige. As you may know, the sanitary conditions at Corinth during that time were deplorable and many men became incapacitated due to illness.
I do not have my research material with me, but I will try to recollect as best as possible. After the evacuation of Corinth, one of the 37th regiments went to Kentucky and the other stayed in Mississippi. The one that went to Kentucky was the one that was later re-designated the 34th -- but I believe the redesignation didn't happen until after the battle of Perryville. The 37th that stayed in Mississippi would later fight in the battle of Corinth in October 1862.
Having said all that, I see at least two possiblitities for the death of your ancestor: 1) he died from an illness he contracted during the seige of Corinth, which doesn't narrow down which "37th" he belonged to, or, 2) he died from wounds he received at the battle of Corinth on October 3-4, 1862 -- which would mean that he belonged to the "37th" that remained designated as the 37th Miss. Inf. Reg.
Hope this helps,