My favorite has been the Mission Multi-Cool/Endura-Cool gaiter.
It's long enough to fully cover your neck, as well as going between your head and the helmet padding.
Material is thin enough to pass a lot of air. Yet, also prevents sunburn. I don't have to wear sunscreen on my neck, even on long summer rides. No sunburn, no jacket collar getting nasty from rubbed-off sunscreen.
You will still want to wash the helmet padding once or even twice per summer, but the gaiter does help a lot.
Buff brand or similar multi use tube scarves
Mission Neck Gaiter Customize Your Coverage, Face Mask, 12+ Ways to Wear https://www.amazon.com/dp/B00UR4HHC8/ref=cm_sw_r_cp_api_glc_fabc_nZFaGbWGY87XB
I don’t particularly like that brand but the photo shows what it looks like. It works very well as forehead coverage but doesn’t get too hot
1) Mission Cooling Neck Gaiter https://www.amazon.com/dp/B00UR4HHC8/
Not appropriate as a face mask, but works really great for cooking you down. I’ve sworn by one for years. I have one right now while I nurse my baby before bed.
2) Ice water foot bath
You don’t realize until you do it how much cooling down your feet will make your whole body feel better.
My family all used neck-gaiters. Their really breathable. The ones with elastic are super easy to keep on your face.
Mission Cooling Neck Gaiter 12+ Ways To Wears, Face Mask, UPF 50, Cools when Wet https://www.amazon.com/dp/B00UR4HHC8/ref=cm_sw_r_cp_api_i_eOKjFbEGBZ6RH
Cooling towel
https://www.amazon.com/dp/B00UR4HHC8/ref=cm_sw_r_cp_apa_i_4l56AbN9DNQ5B
The pith helmet if you can't get your hands on one
https://www.amazon.com/dp/B000GT71IO/ref=cm_sw_r_cp_apa_i_sn56Ab0DZ1BM6)
Badge holder
https://www.amazon.com/dp/B017TH4OLM/ref=cm_sw_r_cp_apa_i_lo56Ab6HF9XA7)
MOLLE Pouch
https://www.amazon.com/dp/B01N9HZR2I/ref=cm_sw_r_cp_apa_i_Zp56Ab3VYFKPF)
Collapsing hand truck/cart
https://www.amazon.com/dp/B00LMH8CCS/ref=cm_sw_r_cp_apa_i_nr56AbNZ6RGQN)
I have one that you can get wet so it's cooling. It's this one. Only $15 on Amazon.
Invest in and use something like this, take some OTC meds like Aleve, Advil, or Tylenol, maybe apply cold pack to back of your neck traveling to MEPS, and wear loose fitting clothes ... do whatever to improvise, adapt, and overcome your natural higher body temperature to cool down just enough get in the door.
I felt similarly but not super severe. Getting a gaiter helped a lot. I recommend this one: Mission Cooling Neck Gaiter 12+... https://www.amazon.com/dp/B00UR4HHC8?ref=ppx_pop_mob_ap_share it’s not bothersome at all!
I use them often. Mission makes a cooling buff and it works well. I have a couple and keep them in the ice chest. Feels great in Texas heat.
https://www.amazon.com/dp/B00UR4HHC8/ref=cm_sw_r_cp_apa_V.eSzbY0GEV1G