I highly recommend that you get a university degree. For many countries, the field matters not, but if you wish to be respected in the TEFL field, you need to have credentials that back up your teaching abilities. You could be the best teacher in the world, but without a degree, you will not be able for advancement in your chosen career field.
In my experience, I have seen many English language training and bilingual schools that will accept you as long as you are a native speaker. If you have any training whatsoever, you would be even more welcome to teach with them. However, you would have visa issues and problems with the local police if your extended stay abroad is not official. Your local school should be able to extend your visa though up to three times and there should be various ways to extend your stay outside of academic reasons.
If you have further questions, feel free to open a secure chat window with me in my on-line school by clicking on the green "Live Help" button.
Eric Paul Monroehttp://www.eric-tesol.com/