I've been there. I'm not familiar with the route you're taking, though. I didn't spend much time on the East side of the divide.
I was there doing trail work, so I cleared trail all the way to the top of the White River Pass, South of Haystack Mountain. Once up at the pass (and at the end of the work day), I climbed up Red Butte to the East for a good view. I pondered the sunset over the Chinese Wall while taking a big dump...
Somehow I didn't get a picture of it (the Chinese Wall) and I can't remember why. Perhaps my battery was dead? Anyway, hiked the 7 miles or so back to camp after dark.
People always consider bears the biggest threat out there but they're wrong. Bears are a threat, and you should be prepared for them. But falling trees are the most dangerous thing out there. There are vast areas where forest fires ripped through in previous years and left all the trees dead. These trees will come down with the slightest breeze after 8-10 years or so. So some will randomly fall without any warning, but whats really dangerous is when a storm rolls through. Huge swaths of them will fall at once.
One day I was clearing a downed tree from the trail when another one fell and missed me by less than 6 inches. It didn't make a sound until it hit the ground. Another day I was clearing trail in another burned area when a really strong storm rolled in. Over the course of about 15 minutes the wind went from a slight breeze to heavy gusts. About the time the tree tops started whistling in the wind, I could see the gusts coming at me by the waves of trees falling. A crewmate and I were out there by ourselves, a mile or so from the rest of the crew. We dropped our tools and ran as fast as we could to the nearest clearing, watching the tree tops behind us to avoid them as they came down on us.
A buddy of mine continued working out there the following seasons. He was killed two years later due to a falling tree.
Don't be fooled, you stand a decent chance of storms in August. Make sure you don't setup camp near any dead trees. Do not try hiking through a burned area in a wind storm. Be aware, August is also fire season. It will be hot and very dry with afternoon lightning storms. A lot of the smaller creeks may be dry by that time as well. As an example, the White River (west of Cliff Mountain) is dangerous to cross in June, but by August it's a trickle.