Jeti-Oguz is workable for most of the year, which makes it one of the more forgiving destinations near Karakol to plan around.
June to early July brings the best wildflowers in the valley meadows, along with strong water flow at the waterfall from snowmelt. Late July through August is warmest and driest, the easiest window for the horseback ride. September brings golden valley colors and thinner crowds, still fully rideable. Winter closes the yurt-camp overnight option but the red rock formations themselves are strikingly photogenic against snow, and the gorge itself stays accessible by road.
If your dates are flexible and you only care about weather odds, late July to early September is the safest bet. If you want the valley at its greenest, aim for June.
Zoomy