I have studied this company for some time. I recommend if you want to the know the history of this company as it pertains to their Captain, Leroy M Nutt, please find a copy of Courageous Journey: The Civil War Journal of Laettia Lafon Ashmore Nutt. This was a journal kept by Capt Nutt's Wife (Her story is no less intriguing living with the army with her children for most of 1864). After the Capture at Ark post of half the men, she basically followed her husband in the Army of Tennessee in which Capt Nutt Served until the end of the war (First as additional company attached to the 10th Texas Infantry serving dismounted until the fall of 1863). In the late fall of 1863 they were remounted and made scouts attached to Granbury's Brigade reporting basically to Col Ben Hill provost martial. They basically performed operations behind enemy lines probably in civilian clothing. It was a very dangerous job.

The history is very difficult to follow because a good portion of the company (as well as other units captured at Arkansas Post) were not captured but escaped and continued to serve west of the Mississippi. Other who were captured were imprisoned in Chicago and then exchanged at City Point Va and attached to Granbury's brigade. those serving west of the Mississippi were eventually merged into other units in the TMD. Nutts command serving East fo the Mississippu basically incorporated elements of the 4 independent LA, AR & TX cavalry companies that were captured at Arkansas Post. It was essentially a mixed command which is why moral in the unit was pretty bad until the winter of 1864. (see 1461 Days in the Confederate Army.)

Many of those who served in the AoT deserted and escaped to go back and serve west of the Mississippi. However a core group stayed and served as scouts once they were remounted and gave up what they deemed as undesirable serving as foot soldiers. Much of the writings of Capt Nutt were attempts to get officially reassigned back the Trans-Mississippi Department. However the group under Nutt continued to serve until the end of the war with the AOT. His rerecords show he served with the AOT in GA, TN and NC and finally paroled in Alabama.

I show in 1964 they were remounted and served as scouts and armed with Mississippi Rifles.

If you want to understand they type of operations they performed please take a look at a paper called "Fagots form the Campfire" by Louis Dupre. http://docsouth.unc.edu/fpn/dupre/menu.html

This is an excellent read regarding scouting operations with the Army of Tennessee.

Sources:

Fourteen hundred and 91 days in the Confederate Army
Courageous Journey: The Civil War Journal of Laettia Lafon Ashmore Nutt
Services Records for Capt. Leroy Moncure Nutt
Also there are some records and papers located in North Carolina as well.