The quote indicates that Maj. Hamilton was at one of his guns when BGen Semmes was taken past them on his way to the rear. I didn't type the full transcription, but the article makes it clear that Hamilton and Semmes spoke for a few moments. With the roar of artillery in the immediate locale and Semmes' condition, this would have necessitated the two officers being within a few feet of each other. It is possible that a Pennsylvania or Massachussetts state flag could be mistaken for a dark blue blanket, but it might also be worth asking our flag experts if a regimental flag would be large enough to serve as a litter, or if it would even support a man.
The quoted letter was dated 26 July 63 - when does the flag story first appear? You may have a primary source reference to this event, in which case I apologize in advance for the rest of this response. Unfortunately there are lots of entertaining tales that emerged post war from veterans that were completely false. Two quick examples:
Brig. Gen. Robert Toombs reportedly found and tended a mortally wounded Fletcher Webster on Chinn Ridge as he rode to rejoin his brigade (it was at Savage's Station, if the teller really stretched the truth, ala Augustus Dickert); copies of the tale seem to have started appearing in the 1890's and spread like weeds. None of my several primary sources support this tale, and based on those sources, I don't think Toombs rode over Chinn Ridge until the battle was over, and Toombs and his men knew their way around the area, having had winter camp nearby. But if you do any kind of historic newspaper work, 1890's on, I bet you'll find this story.
A letter with an unflattering description of Henry Benning at Chickamauga, attributed to Longstreet, appeared in a Philadelphia newspaper in 1875. The letter was reprinted in most major newspapers, in Dortch-Longstreet's book, and other places, including at least one of the modern campaign studies on the battle. Longstreet's description is an entertaining read, but does not dovetail with my primary sources. The letter was angrily rebutted by one of Benning's Colonels, but he was, after all, just a Colonel, and I don't think his letter got reprinted anywhere.
I think we need to be careful with the more colorful, loosely atributed tales that came out of the post war era. I consider them questionable (at best) and would not use them until/unless they make sense with respect to the primary sources.