I had to cancel a trip in late June due to high water flow, so watch the snow levels
Late June through August is monsoon season, so you have to watch the weather and you typically can't complete your reservation until the day before
The flowrate drops as the season goes on so expect deep cold water in June and shallower water in September
During peak temperatures, the canyon stays cool as it sees little sunlight. It has been in the 70's in the canyon and over 100 as you exit.
October can be very cold, I have hiked upstream to the Narrow using a drysuit. Towards the winter season, you may not be able to get a driver to drop you off at the start of the trail.
I highly recommend river shoes and hiking poles. Sometimes it is like walking over alga covered bowling balls. Zion Outdoor Adventure can set you up with everything that you need.