If you want William's records, go here https://catalog.archives.gov/search?q=William%20Braden%20Mississippi

There are only 3 pages, the last one, as I read it dated 1861, that was before Chickamauga. There is nothing about him serving in another unit, however I will try to locate these records for you. If you are looking to place a headstone, you can place a "In Memory Of" as I did for my grandfather he was KIA at Chickamauga.

GP
SCV Camp 2263 "Shieldsboro Rifles"