There is always a possibility that there was a transfer of sorts that has not appeared in any of the surviving records. But as best as I have been able to surmize from the confusion of the reorganization process of the summer of 1862 that Capt Rutherford first organized his company as many others were as an independent partisan command, that was later (maybe just shortly after the battle of Whitney Lane) was assigned to Major Chrisman's Battalion and when Chrisman's Battalion was consolidated with Dobbin's regiment It was still the same company as far as I can tell.

I know that the order to assign Rutherfords Comapny to Chrisman's Battalion is found in Hindman's Copybook of Telegraphs, I just can't pull it up right now. I need to transfer those to a media that I can work with on my computer.

As far as the National Parks Service Civil War Soldiers and Sailors List, as we know, is not all inclusive and is best used as an index and not as a diffinitive source. Especially when dealing with these "early" unit formations of the Trans-Mississippi department in Arkansas