This year is indeed crowded, and unfortunately Great Basin National Park seems to have been 'discovered' in recent years.
I can think of a few alternatives in the same area.
- Mount Moriah immediately north of GBNP. Gandy Road north from the Border Inn has several access roads up the canyons. Hendrys Creek is closest but you have other options as you go north. Check out the hike up to The Table.
- Schell Creek Range is one range over from GBNP to the west. Similar to Mt. Moriah in that the east side has a graded road down in the valley (NV 893 past the wind farm) with accesses branching off up along the creeks. West side near Ely and McGill has a few Forest Service camp grounds along what's called the Success Loop.
- Just over into Utah, Notch Peak might be worth a visit. One of the highest vertical rock faces in the US, like El Capitan. Access from a marked BLM byway off US 50. I remember these mountains as dryer and warmer than the other two.
These areas are all very remote - go prepared and remember safety first. If you go in September, you may want to check when hunting seasons open.
Hope this gives you some ideas and I look forward to seeing a trip report!