If you've got reasonable instructional experience, regional Australia's flight schools may be interested, as not that many instructors make it experienced Grade 1 instructor, those that got enough multi time for airline gigs etc.
You'd obviously need job offer, go through the employer sponsored visa.
Best would be to sort Work and Holiday for US citizens (not more than 30yo when applying), subclass 462 - DIC overview.
Then once CPL/FI conversion done - no set hours for CPL and 10hrs minimum for FI, so long the previous FI training covers the 50hr total -
which may be issue for FAA CFIs, as they'd probably do FI seat flying for 15-20hrs or so, unlike some other countries with mandated minimum hours courses.
I don't think a school would offer you a job without having CASA CPL/FI and knowing you, but what I mentioned above, can help.
I do agree with other posters regarding fresh FI job vacancies and too many local pilots for those.
In a nutshell, too much hassle and cost,
unless you cannot qualify for Australian PR
visa through your non-aviation qualifications/skills/education. Most of them can be found on Skilled Occupation List (SOL) on Department of Immigration & Citizenship
There are other countries where you can go for some instructing without all the hassle you'd have to go through and lots of uncertainties, for Australia, compared to other countries.