The original 13th Texas Infantry had three companies that were raised in Smith County (F, L, and M). Company F was enrolled at Tyler, Texas on October 14, 1861 for 12 months. John M. Douglas was the captain and Jonathan Lewter was the 1st Lieutenant. This company was remustered for the war on March 28, 1862 at Velasco, Texas. This is when Jonathan Lewter became captain. It was mustered as of Speight's regiment even though it was not officially transferred until April 3, 1862. The transfer included companies F, K, and M of the 13th Texas Infantry. Captain Jonathan Lewter's company became Company D, 15th Texas Infantry. Four companies of the 1st (Speight's) Battalion, Texas Infantry were consolidated into three companies and were transferred to the 1st Regiment, Texas Heavy Artillery. The only companies of the 15th Texas Infantry that I have identified with Smith County are D and H, the latter being commanded by Captain John W. Daniel. I have not been able to identify all of the original companies of the 1st (Speight's) Battalion, Texas Infantry. It appears that some of the them were understrength companies or squads.
The flag you are trying to identify probably belonged to Captain John M. Douglas' company later commanded by Captain Jonathan Lewter. It would have been the flag of (1st) Company F, 13th Texas Infantry and Company D, 15th Texas Infantry.