Ausflyer,
For what its worth, I too am an ozzie and live in the US. I have a green card, an FAA ATP and 13500 hours, nearly all jet and almost 5000 PIC wide body command. I have flown for respected major carriers and legacy airlines.
I have had ZERO success in obtaining a job in the US. NOTHING.
I don't have a college degree and I don't even get a reply from Spirit, Virgin, Jetblue etc except for the automated email reply.
I have virtually given up, I don't know how folks get jobs here without someone recommending you but at this stage I will just stick to contract flying in Asia Pacific but the commute is killing me!!!!
Good luck cobber.