Cooling issues

You should also check the General shape of the cooling system. Pump and the shroud have been mentioned. Check the belt condition and tension as well as the coolant level. Check also that the radiator fins are straight and not blocked with insects and other crap.

If the previous owners did not change coolant or flush the system occasionally then the passages in the radiator and engine can be restricted and will not transfer heat effectively . You can get a general idea of the passage condition by removing the rad cap looking down into the radiator the water level is low enough. You may have to drain some coolant out. Are the tube openings clean and clear? If you see deposits restricting or blocking the passages you can try using a flush additive and follow the directions. Then remove the thermostat and reverse flush the system until it runs clear. Check the tubes inside the rad to see if they are less blocked.