Indeed a lot of flags wer etaken at Franklin/Nashville so there would have been a shortage at least for a time.
After this campaign what was left of the AOT fell back into Alabama/Mississippi and was split up. What had been Polk's old Army of Mississippi/Polk's Corps (by this time SD Lee's Corps), was sent back to the Mobile area where many of the untis had come from in May, 1864. These were the units in the 1864 campaign that carried the 12 star Mobile Depot battle flags.
The remains of the organic AOT was sent to North Carolina where it ended the war. As such they had to pass through Georgia to ge there and that was probably through Augusta where the Augusta Depot flags were made. This was the direct rail route to North Carolina from Georgia.
Thus, it is POSSIBLE, that the AOT got new flags while passing through the area. As a related issue, in 1865 the CS War Department issued orders regarding the flags of consolidated regiments, which was going on a lot at this time - what had been 5 original Tennessee regiments, for example, were consolidated into 1 regiment. The WD orders regarded their flags and that only one flag was to be carried by those newly consolidated regiments, so when you factor this in there might not have really been a flag shortage.
This is different from the consolidated units in 1863 where it seems that the regiments still carried both flags - this did indeed happen at Chickamauga, for example, with the combined 15th/37th Tennessee and some others too.