I wonder if this Order that Danny found in the Alexander Papers is a clue to the Original Company H.

6/21 Special Orders #11 Army of the South West
Capt R D Anderson to report with his Co to Col A W Johnson at Trenton
The conscripts brought from Phillips county by A C Robertson and now attached to Col A W Johnson will constitute a Co under John Horner as Capt

and also:

6/26 Hindman to Johnson-I telegraphed you that I desired the conscripts who were enrolled by Col. A. C. Robertson, some sixty (60) or Seventy (70) in number to be formed into one Co under command of Jon J. Horner as Capt, Robert T. Shell as 1st Lt, the two 2nd Lt’s to be elected by the men.
Lt Shell has reported here in person, and had not heard of the telegraph to you on the subject. I therefore send him to you with this letter, and wish that he shall organize and take command of the Co, until such time as Capt Horner and those with him at DeVall’s Bluff can be spared from their present duty. The Co will of course be of your command.