The 13th Illinois lost 2 flags at Chickasaw Bayou. They almost lost another flag at the battle of Ringgold Gap, Georgia. The color bearer, Corporal Sackett had advanced with his regiment, he looked back and realized that only he had made it the farthest. He got down and put the flag on a tree branch about 60 yards away from Govan's position. Govan's officers begged to go down there and get that flag, but Cleburne said no.
I did not see any 13th Illinois flags recaptured at the National Archives, War Department list.
Good luck on the find!