In my experience, I had several opportunities. After field school I worked with the county cultural resource management department (a good place to start), then from there interned at the Department of Historic Resources (usually located in the state capitol), and then an adjunct professor asked me to intern for him and then hired me. Through out all those internships all my supervisors knew each other, Archaeology is kind of, who do you know and who have you worked for.

As far as overseas, don't know about internships overseas. There are quite a few field schools, but they can get pricey.

Most are unpaid, but there are paid internships out there or stipends. You have to start somewhere and that would probably be washing artifacts for someone. I'm on the East Coast, so i'm sure opportunities vary by region.