Hi Justine, I'm surprised to hear about the referees being an issue with the agencies. One thing that employment and temping agencies are used to within Oz and that's migrants registering for work. To tell you that they can't register you due to a lack of Australian referees is worrying and my recommendation would be to just avoid those that say such a thing.
In my expereince in this industry, the referees are contacted via email directly by the agencies anyway as part of the checks.
Lily Rainbow is spot on with her suggestion of doing some volunteer work if you are in a position to do that. Adelaide is definitely a place where 'who you know' counts. Particularly in sectors such as Admin and HR. Also, ensure your Resume is not putting them off. I'd be happy to give it a lookover for you if you wish.
Good luck and, if there is one piece of advice I could give you, it is be persistent! It can be a slog getting work here in certain sectors but just don't give up. Once you have your foot in the job market door, you're pretty much sorted.