Saludos Harry

I can't really help with South America, but in Central America there are many many jobs available for native speaker teachers who are actually there. The majority of schools don't advertise on internet, but in local papers or just via word-of-mouth. I imagine the same thing applies in most of South America.

In Central America pay is very low, but so is the cost of living. It is a great place to get some teaching hours under your belt, and almost any job will open the door to better jobs later on.

Your best bet, if you can afford it, is just pack your bag and go.