Memphis was under Union control after June 1862. Most Civil War dead were buried in Elmwood Cemetery (both Umion and Confederate). The Confederates are still there in "Confederate Rest" and the Union soldiers were moved to the National Cemetery when it was later established.

Elmwood has extensive records, but I don't think any are available on line. The staff is usually very helpful to researchers and you can find them on the web at
http://www.elmwoodcemetery.org/main/index.htm
Postal address: 824 S. Dudley Street. Memphis, TN 38104
Phone: 901-774-3212 · Fax: 901-774-0085

The original National Cemetery at Memphis is "full" and so there are now two of them; the old and the new.
The link below has a directory of names for the old cemetery.
http://www.interment.net/data/us/tn/shelby/memnat/

I very enjoy reseaching Memphis history but have to admit this was the first mention of the Overton Hospital I have come across. A quick search showed that the buidling was orginally built as a fine hotel and was taken for a hospital by the Confederates and remained in use after the Union occupied the city in June 1862. One of the founders of Memphis was Judge John Overton(of Nashville) and he may have owned the hotel. His son built the famous Maxwell House Hotel in Nashville and after the War his son-in-law, Robert Brinkley built the original Peabody Hotel (1869), forerunner to the present Peabody Hotel (1923); famous around the world for the Ducks in the Lobby Fountain.

A description of the Overton Hotel (Hospital) I found said it was larger and grander than the Gayoso Hotel (which was pretty large and grand!) The Gayoso was occupied by many Union officers during the War. (My great grandparents were later married in that Hotel; then it burned in 1899).

Hope you find him!