I'm guessing that you have a work visa, not a working holiday one.
If you have S92 experience hit up HNZ.
CHC, were looking for some experienced and type rated casual offshore pilots just recently. It may be worth calling them.
No Idea about Bristow.
Babcock is mostly word of mouth, however to stand a chance in Australia at the moment expect to be ATPL/IR qualified. There are heaps of ATPL/IR qualified pilots in Australia with those quals that are looking.
Good luck and welcome to Australia.