From my notes on the 15th Texas Infantry, I find that they left Texas [along with many other newly formed Texas regiments] to join Hindman in Arkansas.
Tyler, Texas was the principle rendezvous point for many of these regiments, including the 15th. The 15th left Camp Crockett near Millican on June 29th, 1862 and arrived at Camp Daniel in Tyler on July 29th.
Like many of the other regiments, they remained in Tyler for some time drilling, arming and in general, outfitting themselves for the Arkansas expedition. The 15th Texas remained in Tyler until September 9th, when they took up the march for Little Rock.
The citizens of Tyler were very hospitable as indicated in this article from the August 21st edition of the Tyler Reporter:
On August 13th, we attended, by special invitation, a barbecue given to Col Speight's Regt by the citizens in the vicinity of their encampment, twelve or fourteen miles North-west of this place. We arrived on the ground about 9 o'clock in the morning, & found the troops already on parade. The Regt was reviewed by Gen McCulloch, who pronounced the troops well drilled. Gen McCulloch addressed the Regt in a stirring appeal, the good effects of which were visible. Dinner time arrived, & with it a bountiful supply of such substantials as are customary on such occasions. A dinner at a private house could not have passed off more agreeably. After dinner, the troops & citizens were again assembled, & Col Speight in a chaste & neat address, returned the thanks of his command to the citizens for the honor done them. Maj John Henry Brown, of Gen McCulloch's Staff, being then loudly called, indulged in a short, pointed address—full of humor, good sense & excellent advice. Lt Col Harrison, being called, also indulged in a few pertinent remarks. The whole affair passed off well, & we believe all present enjoyed themselves, notwithstanding the intense heat of the day. Speight's Regt enjoys an excellent reputation for good behavior & the exhibition of all those characteristics which to go make the good soldier.
It could very well be that the flag was presented to the Regiment by the Tyler residents during their length stay near town.