Trinity Alps half as far away
It doesn't look like there's a lot of long hikes for multi day backpacks in Great Basin. There are unlimited possibilities in TA
I bet Great Basin gets less snow so maybe accessible a little earlier in season
This forum is used to share your experiences out on the trails.
The 3-day backpack trip I did is about it unless you can do some much rougher terrain. Starting the trip with the dayhike of Wheeler Peak adds some car camping time too if you like to mic car camping and backpacking on the same trip. TA has far more terrain for longer trips, but GBNP does melt out earlier it seems being so far inland and in a desert. I basically did the whole park except the cave tour this trip. There is a large arch in the southern part of the park but the area was closed now. In theory you could pack down there. It's four hours from SLC.
My trips at http://www.deanmyerson.org/photoindex