Whether or not there were actually field medics isn't all that important (although apparently there were--bonus for you!). Historical accuracy is nice because it helps you put together an outfit that has internal consistency, but you can easily get that without it, especially if you have a concept in mind. Like you do!
So. Civil War uniforms are a great place to start and there are heaps of resources for that. McCall even has a costume pattern
out right now that would work well. You can have a lot of fun with the insignia, if you're inclined that way--does your sergeant get a certain number of bars, or chevrons, or pips, or does rank follow a more Southern scheme and show in how much and how complex the trim on the sleeves? What's the symbol for "medic" in his world? Has he earned medals for merit? (I love making up medals so much.) Does his particular company have a crest or a color? Oh MAN, the possibilities of just the uniform's trim. I nerd out on that stuff like crazy, though, so don't get the idea you have to create an entire world to justify some gold braid. You don't have to. But you CAN!
Anybody doing work in the field can also carry around all KINDS of stuff. Get a nice sturdy belt so you can attach things to it. I think a really cool thing would be a pouch or bandolier of corked vials, all sorts of fun powders and potions with aged labels (or no labels at all, patient beware). What you do for gadgetry can depend on the kind of medicine you want your guy to practice. Have mad scientists in his world advanced the frontiers of medicine to the point where he might, for example, have some kind of arrangement for re-attaching limbs?
The sky's really the limit here, and you can also keep things pretty simple if you want (i.e. "This bottle makes you hurt less if you're gonna live, and this bottle makes you care less that you're gonna die"). A doctor's bag should do you as a basic thing, and those are fairly easy to find in antique stores. It'll be handy for carrying around normal things like a cellphone too. A leather or heavy canvas satchel would do the same.