My cat appears to overheat during a day in the outdoors. What can I do to help him stay refreshed and stay cooled down?
No need to refresh your cat, he will be perfectly capable of staying inactive for longer periods of time during the hot hours of the day. They are genetically programmed predators that are good at conserving their energy and finding secluded spots, lurking on their prey in ambush when conditions are too hot and they might exhaust themselves and overheat too fast.
Make sure your cat has a shade to retreat to and access to fresh water. The place it stays at should be well ventilated, with fresh air coming into it, preferably with access to the outdoors, if at all possible. Cats will often retreat into the bushes or high grass during the hottest hours of the day, but it's also perfectly normal for them to enjoy a bit of the sun too. Don't worry about their fur getting too hot, it's well insulated and with most cat species, it will take a very long time for that heat to propagate onto their body. Cat species with shorter fur will only have this heat tolerance threshold ever so slightly lower than longer fur ones.
They also exchange the heat mostly through breathing and pads, and not through their fur, so perhaps giving your cat access to areas laid with floor tiles or any other cooler surfaces wouldn't be a bad idea either. In short though, as long as your cat has all the essentials provided for, like access to fresh water, shade and well aired areas, they will do all the refreshing on their own well enough even in the hottest of climates during the hottest of hours.
Do not use however some water sprinkler or a spray bottle on them with intention of helping them cool down. Cats utterly dislike that and when they learn what to run away from, they will and you'll only achieve the opposite with it, stressing and overheating your cat further.
I live in the Caribbean so it's pretty much always hot here and whenever my cat gets restless (constant kneading, meowing, clawing etc). I give him some cool tap water and anytime the fan is on, he likes to sit on the back of my chair apparently enjoying the breeze.
He also seems to like the cold floor tiles in the day (I rarely see him on the rug) and at nights I put a hard pillow in his basket and keep it next to a slightly opened window.
So, other things I do are to keep curtains closed so that the direct sun can't reach him when I'm not home but keep doors and windows open a bit for the occasional breeze to pass through and also keep his water in an open container in a corner.