Is it possible to add a file to an existing branch?

Say you have a large assembly and want to explore a change.  You branch it and a couple of component parts.  Later, you realize you need to look at changing an additional part and want to add it to the branch.

Are you limited to creating a new branch name for that part?  Soon you can have a mess of branches associated with your project.

I think the answer is "no", but perhaps there's something hidden away I'm not seeing.

Also, is there a way to see a list of all the branches?