Which is probably why he said he's not looking at a pilot-based work visa.
The fact that you have a degree is irrelevant; what you need to get work as an instructor is an Australian CPL and Instructor Rating. Your prior teaching experience will make the instructor rating course easier than someone with no teaching experience.
If you want to fly multi-turbine-charter, though, forget instructing. Go north, fly a 210 on charter, and progress that way.
Schools who do instructor ratings often (but not always) hire their own graduates, so discuss this with your potential schools before you part with any of your own money. It is, however, not at all a requirement that you trained with someone before they hire you.