Bette
A family researcher at this site FamilySearch.org has Franklin Houston Coney's death date as July 12, 1863 in Clinton, East Feliciana Parish, LA.
There were several hospitals located in Clinton treating CSA soldiers. If the above date is correct, he is likely among the unknowns buried at this site.
http://www.geocities.com/ccca70722/
In order to verify this data you will need to order his military service records, the banner at the top of this page provides this service.
Mitchell