Drew,
I lean towards a new flag coming from Mobile. As I said, Hood's army drew its supplies from Alabama and Mississippi including from Mobile. Some came from Georgia, mainly Macon and Columbus, but the rest were from Alabama and Mississippi.
As such, with Mobile being the major source for flags for that department, replacement flags were quite possible.
Sadly we may never know the ID of the flag taken at Franklin since it is unmarked barring a letter or diary where the author mentions his unit losing a flag from one of the regiments of Sears' Brigade.
Greg Biggs