I thought the older 4L80s only locked up in 4th gear, but some reading suggests that they also lock up at higher speeds in 3rd gear.
There are many relatively inexpensive lockup kits that allow lockup in 2nd, lower speed lockup in 3rd and 4th gears. Worth looking into, IMO.
That said, I don't believe that trans should have gotten near that hot towing what you towed in temperate weather.
Remember, they were originally designed to tow more than that without aftermarket parts.
Your new to you has 160k of unknown miles and treatment I'm guesisng.
Could be and likely is any host of issues that are contributing to trans heat. Cooler clean? fluid age/quality, bad lockup switch, worn converter, trans slipping a bit?
IMO, the primary issue is that something isn't 100% with the trans.
A cooler, a deeper pan, different driving habits may over come that, but you should be able to keep it cool in 3rd gear going up a hill.
(Disclaimer, I don't know the grades you were climbing, but presumably not "that" onerous if you've run the same older truck over it before without issue. It has a trans temp idiot light too.)
