I've got work lined up in the USA flying a SAR helicopter, though I'll be a green card holder (wife is American).
A buddy of mine works for the same company on an E3 visa flying a firebombing Huey and b212 and he has no degree and basically comes over for the fire season. Took him about 2 weeks to get the E3 in Melbourne, he had his FAA ATP a few years prior to applying for work if the E3.
It's all doable on an E3 but most US companies don't understand the E3 visa and as result confuse it with sponsorship. If you have an Aussie friend over there to grease the wheels and explain it, makes getting a job offer easier.
Don't expect any offers unless you have all your FAA tickets in hand and someone to explain his the E3 actually works. And it takes a while to organize going through all the TSA stuff to convert over to an FAA ATP.