A trick a friend of mine pulled at least once (though i have to say she did eventually have to leave due to no permanent visa) is to fly out to New Zealand on a visit before your australian visa expires and then fly back into the country and apply for a new visa as you land. It may work still, not sure if they have changed the rules or not in the past 10 years.

If you are serious about staying in the country then I would look into acquiring a permanent citizenship. However, you will need skills that are needed in Australia at the moment (things like teaching, police, doctors etc are pretty much a dead cert but other careers less likely) and you may have to return to the UK to apply. As suggested above, talk to an immigration officer and read up on the current laws.