I guess my best chances as a non-native speaker are to get a specialist visa to work at eikawas.
I've taught at high schools here (I did about everything there is here, from EP first graders to private lessons for adults) and am not too fond of big classes of unruly teenagers, especially if I am just an ALT who is something of a prop.
In 1990 I went to study Japanese in the morning, which I can only recommend to everybody, and taught at private schools in the afternoons; I also squeezed in some karate training and went to auditions. The second year I got a student visa again but had so many lessons lined up that I didn't have time for school. Hopefully things haven't changed too much!
Since I have a TEFL (and more than 12 years of experience) I guess there's no advantage of getting a TOEIC or an IELTS if they don't improve my chances of getting hired.