Hey Stretch,
I've flown east west a few times, but never from here over to there.....but I guess the principles are the same, possibly better with the winds going to the east if you can get amongst it..!
First port of call would be Kalgoorlie.
From there, depending on your comfortable endurance, it's either north east thru Warburton to Ayres Rock to Alice. After that, I've never been past Alice.
Other option from Kal would be thru Forrest, Ceduna, Broken Hill, then come up towards your destination.... Probably a bit longer way, but potentially more places and services for fuel and accom etc....
I'm sure others have gone straight thru the guts and have some better info for you...!