Like so many things down here, there is a "fee" associated with extending your visa. It's not too bad, usually $100 or less. The other option, which has been posted, is to head to Belize. You can do that with Vern on his Catamaran. All in all, most instructors have to deal with this on a regular basis. They are also on the extended stay program. There is no real solution as of yet for those who want to work here, other than applying for residency. That is also another ordeal and well beyond the scope of what you seem to want.
Best of luck,
-marco