OP, it's my educated guess there's absolutely nothing wrong with your trans cooling system. The problem stems from you using 4Hi for a long extended climb. I monitor my trans temp via a Bluetooth reader sending info to my Torque Pro app on my Android phone. I recently took my rig wheeling with [MENTION=148218]JZiggy[/MENTION] (aka Jordan) in the hills above the California Coast near Big Sur. The climbs were steep and sustained. At first, I had my rig in 2Hi because the terrain was mellow and didn't think I needed 4Lo. I was watching my trans temp rise, and when it got to around 230, I pulled over to let my trans cool down. We started off again and the temps spiked again. Jordan asked me if I was in 4Hi or 4Lo. I said 4Hi and he suggested I try 4Lo. Man, what a difference. After switching to 4Lo, my trans temps never got above 170.
This was an important lesson to learn about 4wheeling with an automatic transmission. I never knew the difference in the trans temp would be so dramatic between 4Hi and 4Lo. My previous 4wheeling experience had been with a manual transmission. There's probably a lot of people who've overheated their automatic transmission while 4wheeling but they didn't necessarily know it because the "idiot" light didn't light up on the dash. I think these dash lights were affectionately named "Idiot Lights" because the light is telling you, "Hey Idiot, you've got a problem. Stop!" Like others have mentioned, the light comes on around 300 degrees and that basically means you should stop and not drive anymore because your fluid is cooked.
So, #1, it's important to monitor your trans temp. If you have a 96-98 and can't pull trans temp info via a scan gauge or Bluetooth reader, install an aftermarket gauge. If you have a 99-02, definitely invest in either a scan gauge tool or Bluetooth reader sending info to an app like Torque Pro. #2, use 4Lo when climbing sustained grades off-road. Your transmission will thank you.