Great thoughts to share your loss in a thread. Very timely, as well. I’m sorry you lost a chicken, though.
I have had exceptionally good results using misters. The ones I use are free standing and attach to a garden hose. They can be easily be moved. I start them up late in the morning, say around 11, run them for about 30 minutes, then turn them off. I don’t need to soak the chickens, but I do want the area around them, the landscape fabric shades over the run, and yes, even the litter somewhat, to be wet. For the next couple of hours, as that water evaporates, it cools the air around the run. Then as the day heats up, I run them again periodically, moving them so they are always in the shade, but since the chickens and their area got that initial cool-off between 11 and 1 or 2 (depending on how got it was that day) they seem to stay fairly comfortable. Their run door is open all day long, so they tend to forage in the morning, usually staying in the shady spots and moving with it. As the day wears on and the misters go on, they usually gravitate back toward the run.
Another thing I do is the usual frozen watermelon/fruits/and vegetables, but I stuff them in suet cages and hang them up just at the level where they can reach them.