I visited the Confederate Research Center at Hill College where I found where the Confederate headquarters made a mistake giving the numerical designation of the 1st Texas Cavalry the 31st Texas Cavalry without proper authority. It seems it was merely a clerical error.

This is what I found at Hillsboro:

"After the regiments of the Arizona Brigade were assigned to different brigades, this regiment was known in the field as "Hardeman's" and as "31st Reg't Texas Cav." - No offical designation or recognition of this reg't appears to have been given by General Headquarters - another reg't have being designated as 31st Texas Cav."

Regimental Returns on File
"Hardemans Reg't Texas Cavalry - July 1863
"31st Reg't Texas Cavalry (Col. Hardeman's) - March 1865.

A hand written note found in the archives:

"The organization that has been indexed as the 31st (Hardeman's) Reg't Texas Cavalry first served as the 1st Battalion Arizona Brigade, afterwards Hardeman's Reg't Texas Cav. It then assumed this numerical designation (31) apparently without proper authority. The latest record on file, roster dated May 11, 1866 which is also signed by Peter Hardeman, shows it was finally designated Hardeman's Reg't. In view of the fact that there was another organization which appears to have been properly designated the 31st Reg't Texas Cav., also no evidence has been found to show this organization was ever officially reorganized or given a designation by the (headquarters)."