Well I would check your thermostat. It might be stuck in a more closed position that is restricting proper water flow.
usually when I redo a cooling system in a car, whether I change the radiator, water pump, or just the coolant, I usually go for a thermostat change if its running hot.
You could remove yours and put it in some boiling water I believe, and see if ti opens fully or not. ITs not hard to replace the thermostat, and I'm sure it is not that expensive at all on this site, with the gasket and sealant if necessary.
If its not the thermostat, you should make sure your fans are coming on, as it might be the sensor for the electric fan, that tells the fans to turn on at a certain temp.
If the fans are ok and thermostat is ok, then maybe the only thing left would be the water pump, or maybe the fan clutch for the engine fan if your car is so equipped.
So far my car has yet to break the 90* mark with AC on full in the sun at about 85-100 degrees outside. (varies with where I have been the last few days. )
I remember when my jeep would run hot, I took it to the jeep dealer, they replaced my water resevoir. I told them change the thermostat, they didn't. I took the car back still running hot. I told them to change the thermostat. They did not want to, I literally had to force them to change it, they kept saying it can't be the thermostat.
Well to make a long story short, the thermostat was changed, and the car has not run hot for the last 3 years, and I never went to that dealer again.
So I am thinking that might be your best bet.
'92 300CE - Sold
2004 C240 - C7 Wheels
2015 ML350 - P1, Pano, Ash poplar wood, Sport Dinamica interior, Running Boards Keyless go.