The Official Records will include reports and communications that name the regiment. Don't look for company level data or you may never find anything.
The best approach for this kind of study would be to learn who Woodward reported to and follow that officer's name in the records. For instance, Woodward's regiment was involved in Forrest's Christmas raid into West Tennessee. You really have to read Forrest's report or find a book like Wyeth's That Devil Forrest to understand what was happening at the time and how your ancestor fit into the picture.