Good luck with your screenplay. I'm mainly bumping your topic. However, I imagine a person traveling around would not use a bank much at all unless they had someone in a fixed location to help them.
Why? Because up until pretty recently (as in, I'm in my 50s and I remember this), you couldn't really effectively bank across state lines. Hence I suspect your traveling bounty hunter would take his/her pay in something like gold or silver or would send the payment draft to the home base of operations. This payment draft (a bank draft) is basically what we would call a cashier's check. See: http://www.accountingtools.com/definition-bank-draft
Or they could, I suppose, be paid in a deed to land. But using a bank in Tennessee would probably be hard to do if you were in Michigan. Even if the banks communicated and worked together well, the post still took several days and could be subject to theft.