Stephen F. Mitchell was a member of Company K (the Choctaw Rebels)which was raised in Choctaw County.
A capsule history of the 24th. Regiment can be found at this URL.
The family researcher at this URL gives Pvt. Mitchell's death as 1862 in Perryville, KY.
His service records should help indicate when and where he died. These records can be ordered from this site by clicking on the banner at the top of this page.
Hope this helps,