Engine compartment cooling

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You really can't make that correlation between under hood temps and it causing your engine to run hot. That is, unless your engine compartment is so tight that your puller fan doesn't have enough power to push the air out from the engine compartment. I doubt if that's the case. A "near overheating" engine will cause the engine compartment to run unusually hot, not the reverse.

There's plenty of possibilities for a too hot coolant during idle problem:

Water pump/crank pulley ratio off, under driving the water pump. Aftermarket pulley manufacturers (March included) like to under drive the accessories with their kits, including the water pump. 1 to 1 is an ideal ratio.

Under voltage condition to fan caused by an inadequate wiring/alternator system. A lot of wiring harnesses do not have an adequately sized fan circuit for today's hi amp monsters out there. I run #8 stranded for my fan circuit and run the Bosch 60 amp screw lug relays. My main feed wire to the alternator and battery is 0. For what it's worth with my Painless wiring harness I had a whopping 10.6 volts at idle making it to my fan and the alternator output voltage dropping to 11.6 volts, with the lights off. Problem solved, with a one-wire 180 amp alternator and new hi-amp wiring. Now have 14.2 volts at the alternator and fan with the motor idling and every electric/electronic device installed on the car running. The fan now spins noticably faster.

If you don't already have a shrouded inlet (in front of the radiator) then there's a potential problem. At a stop the "hot air" pressurized engine compartment has a tendency to recirculate the air to the front of the radiator.

Run a Mr. Gasket/Robertshaw performance thermostat. It has a larger open "free area" which, all thing being equal, flows more coolant than a standard thermostat.

Its an easy check on the electrics and pulley ratios.

Good luck on the fix...