Approximately 160 km from the highway to Uluru is Kings Canyon. If you have time, you can easily spend a day there. There is a campground, where you can stay, but you can also spend a night somewhere just by the road.
Camping at the Picnic stations is prohibited. But they are not really checking on that – just get up early enough (according to our experience before 8 o’clock in the morning). You can check out the place in the map below.
There is nothing there, so you need to carry food and water. There is a small shop though at the campground.