The sooner the better, as far as warmer weather goes. June, July and August would be scorching hot, but doable if you like to cook yourself medium well. To me, the best two months are either April or October, with the latter being better due to a little longer days. Temp wise, you should be fine from March-mid May and Sept through the end of October. Any later in the year you can get snow. You can also get snow in March, but it usually melts quick.
Rock rails are pretty much needed. A winch is very helpful. More than one vehicle is highly recommended. Many spots deserve caution and you will be asking yourself, "what was I thinking." And you'll consider turning around. With time, caution, and assistance from a second vehicle, it is doable. It will take time. There will be upwards of 35 miles, one way, of slow crawling. Up on Grey Mesa you can step on the gas for some refreshing dusty two track, but on either side, slow crawling is the order of the day. You cannot get all the way to Lake Powell, but you can see it from the end point. You can hike close to Lake Powell if so inclined. Challenging named obstacles are: The Twist, The Chute, climbing up Grey's Mesa (the Dugway). You get them all again on the return trip back out. You can also take a side trip to Lake Canyon from either end. Until about 10 years ago you traversed Lake Canyon. It washed out and you have to go the long way around now, the bypass not actually on the Hole in the Rock trail itself.
I found this guide to be very helpful and used it. (I printed a copy to have with me). https://www.traildamage.com/trails/index.php?id=100
I'd love to join with you guys. I love the history of this trail. But I cannot get the time off. I'm saving up for a big two week trip in June. If you have time, stop in at the Bluff Fort in Bluff, Utah for some history about the area and the trail. I have several local Salt Lake friends whose families were some of the original travelers of the trail back in 1878 and 1879. The trip was supposed to take them 6 weeks; but it took them six months.
2015 Cherokee Trailhawk:
Granite Crystal, 3.2L V6, 8.4 Nav, black leather, sunroof, tow, tech, comfort/conv, no hood decal, 9 speakers, CD player, Maximum Care lifetime warranty, Mopar first aid kit, rear rubber cargo mat, Thule Aeroblade cross bars (black), Thule Rapid Aero feet, Kuat Vagabond X cargo basket, RRO rock rails with smooth covers, Mopar door sills, Raingler ceiling net, Raingler seat headrest handles, MFC aluminum lift kit, RRO winch bar and Warn M8000S winch, BFG T/A KO2s