Shade. Carry an umbrella, and start early. Here in Florida, people will be using them for the course walk and driver's meeting. Start early.
I keep the top up on my Miata between runs.
Water. Drink a lot, preferably cold, but even if it's not cold it's going to help.
If you get to the end of the day and haven't had to urinate at least a few times, you're not keeping up.
Clothing. I wear long sleeve Columbia brand fishing shirts with the vented back. There are lots of knock-off brands that are less expensive and still comfortable. They are made for outdoors use in hot weather.
The big hat is a meme for autocrossers for a reason. I just bought a new one, and it's about the stupidest looking thing I've ever put on my head. If I was at home I would post a picture of it. Perfect.
I'm going to try out a switch to long pants this summer, the same light weight, hot weather fishing style as the shirts. I've tried that out the last couple of events, and I found that I was cooler than when wearing shorts.
This will eliminate the problem of getting sunscreen all over the car. I'll only need to put it on my face and neck, and the back of my hands.
Evaporative cooling. I wet down my shirt. The quality fishing shirts are made to promote evaporation. the knock-off brands may not be as effective at this.
If I'm going to be getting in the car, I'll wet the sleeves, shoulders, and the front of the shirt. If I'm out working on the course, I'll also pour water down the back of the shirt.
You can wet a bandana and tie it around the back of your neck, or preferably use one of the products that's made for holding water and promoting evaporative cooling.
Acclimation. Particularly if you work indoors, it's going to be harder for you to adjust to being out in the heat. It gets easier, the more you are exposed to it.
Finally, my helmet is white for a reason.