Alan,
I am by no means an expert on the unit, so if I have overstepped my boundries, I sincerely apologize.
I know the men of Co. C, entered service either July 12, or 21 Aug 1862. The last roll I have for them is 28 Feb, 1863.
I need to dig my letters out because I believe they didn't disband until Vicksburg was basically surrounded by Grant. Also I need to dig thru them as I had in my mind that they were only 6 month troops too, but I know I have letters from April or so. I don't remember anything in that set of a reorganization (set could be incomplete) Maybe these two collections I have found will shed some light.
On a side note, one of the cemeteries I cut today is the final resting place of my ancestor who was in Quinn's, Micajah Wilkinson. (my gggreat gandfather) After the disbandment he served in Co B, Powers' Cavalry Regiment.

Seems to be alot of questions on Quinn's in the last few months!