There are references to coats being issued to soldiers. For instance Oxford regiments were given coats, breeches and montero caps in 1643, so if you were an Oxford royalist in July of that year you would wear a nice bright red or blue coat, though not necessarily a well fitting one as uniform issue clothes have never been made to fit.
If you were part of a militia or a trayned band you may have had an issue of coats or you may just have worn your own clothes. In this case your top layer would either be a plain Soldier's coat or perhaps a civillian tailored doublet.
Thus the big problem is that civil war soldiers were not uniformed or regimented in their dress in a way that modern troops are, and as I have described, the coats varied widely even from season to season, so for a particular company or group in reenactment you have to decide the period you are presenting and how to portray the effect that campaigning would have had on the clothes whilst presenting some kind of unified whole.