If you can, it’s advisable to get your work visa before arriving in the country, because sometimes they will make you go to the consulate to fill out some paperwork. Like many other Latin American nations, paperwork and red tape is inevitable, and the process can be long and painful. Any long term visa will probably have to be renewed after a year.
The cultural visa, while a window into the country for volunteers, is not limited to them. Certain institutions will arrange them for you even if you came in on a tourist visa. And if you don’t have an institution to back you up, it is possible in theory get an Ecuadorian citizen to sponsor you.This is obviously the most difficult thing to do because not only do you have to know someone well enough to do so, but they have to put themselves up with some collateral as well.
Otherwise, you could try going through the process of leaving the country every three months, though as stated before, it is becoming increasingly difficult. And that’s not just Ecuador. Many countries in South America are starting to crack down on visa runs.
Check this page on Ecuador visas on the Ecuador embassy in Washington D.C.’s website. Visa rules change so be sure to go to official government sites to get the latest information.
TEFL English teaching job listings in Ecuador
A general overview of what to expect teaching English in Ecuador
Detailed living costs for teaching English in Ecuador