I see you already have a new radiator so assuming it is up to specifications for a vehicle with aircon and even more so if automatic gearbox as well, then that should be OK as it would have been my first thought if only running hotter on steep hills and no loss of coolant.
The danger with an engine running hotter is not (within reason) the actual temperature as long as the coolant pressure cap is in place and working correctly, but when pressure cap is released or not working correctly then the water boils , once it starts boiling it becomes a gas and causes local hot spots inside the engine as it boils at much lower them easily and damages the engine.
This is in the same way a pressure cooker works.
When I was trained as an apprentice in the 1960s I was told for every 1 pound (1lb) of pressure on the radiator cap the engine water would boil 1.5 degrees Centigrade higher, so if water boils at 100 degrees Centigrade at sea level, then a working 10lb pressure cap would allow the water to boil at 115 degrees Centigrade.
The danger is when someone opens the radiator cap on a hot engine and instantly the water boils , shooting hot water everywhere!
Many vehicle radiator/coolant caps are marked 2Bar these days which means 2 x Barometric pressure of 14.7 pounds at sea level (roughly 30 pounds /psi) so giving a fair safety margin as long as all hoses /radiator/cap etc are in good condition.
When engine does run hot and you pull over and stop , it is probably better to leave the engine gently idling if no other problems as this allows the water to circulate around the engine and radiator allowing it to cool down gentle, if engine is switched off the cylinder head/gasket area can actually get hotter.
It mat be worth getting an after market electric fan or switching that you can control if you are concerned.
Sorry if I am "over explaining" I tend to according to my daughters.