In order to get rides North or West (Alice Springs or Adelaide), you need to get to the A1.
There are 2 main options - 1. There's a big truckstop / petrol station a few km out of town, just after the caravan park, where lots of trucks stop. It would be a good place to ask the drivers and catch a ride.
Asking for a ride to Port Augusta might be a good idea. I waited in that station for about 15 minutes and got a ride to Port Augusta and from there a ride all the way to Perth.
2. You can also stand on a random bus stop or other place along the A1 and try to hitch from there. You can get there with bus number 224 or 225 from King William st. in town. But try to avoid that if possible, it can get hot, there's no shelter and people are not keen on stopping.