Something similar happened to me on my '70 'vert (318, non-AC, 19" radiator -- yes, it is small but that's not the issue).
After I bought the car in March 2018, I had the water pump/timing chain replaced and the original radiator recored. All new hoses, new thermostat. Generally ran warm but not hot for the first 15 months, EXCEPT that the temp needle would reach the upper (i.e., RHS) end of the "safe range" when in stop-and-go traffic or idling for more than a few minutes in hot weather. I was not worried as temps would drop back down once the car started moving. Then, my car overheated at highway speeds on the way back from Carlisle this past July and I had to limp back home (drive 10 miles then cool for 15-20, you see the fun).
I thought the original 4-blade no-shroud fan was the issue. I replaced it with a 6-blade shroud fan from a formal Fury (it fit right in, with minor inserts added to anchor the fan without messing the original radiator). I replaced the t-stat for good measure. The engine felt smooth, but it was still overheating AND one could see bubbles in the coolant when running the engine with the radiator cap off. One top-end rebuild and a thorough cleaning later, the car is once again great. Temps are at the very bottom (LHS) of the normal range and, in 95 degree weather stop and go, did not even reach the middle of the range.
Short story: the main issue was the head gasket PLUS the fact that a lot of carbon buildup (probably from lots of small trips made by the first two owners in the first 15 years of the car's life, as
@cbarge ventured) started getting loose and clogging the cooling. Hope you are luckier than I was. I again love my car.