I admit to confusion regarding the 31st Texas and the 1st Texas Arizona brigade regiments. As with other Texas cavalry regiments, it seems that the same number [31 in this case] was used by both regiments, since the 1st Texas of the Arizona brigade was sometimes referred to as the 31st Texas. But, Hawpe and Hardeman commanded 2 separate regiments.

It is my understanding that Hawpe's 31st Texas was part of the brigade that ultimately became Polignac's, and had nothing to do with the 1st Texas of the Arizona brigade [aka 31st Texas]. In fact, Hawpe's 31st Texas served as dismounted cavalry for most of the war.

So I do not understand your comments:
When the 1st Texas Cavalry regiment arrived in Arkansas, Maj. Michael Looscan assumed command of both Col. Hawpe's 31st Texas Cavalry and the 1st Texas Cavalry.

After the battle at Cabin Creek on September 19, 1864, Maj. Michael Looscan's 31st Texas Cavalry and Col. Hardeman's 1st Texas Cavalry and the other Texas regiments returned to Fannin and Grayson County in November 1865

I'm sure you meant November 1864, but regardless, I do not believe these 2 separate units were travelling anywhere together.

Major Looscan may be the only connection between these 2 units. For some reason, he is sometimes listed as Major in both units, further adding to the confusion.