Most likely the coolant was pushed out when the cap wasn't seated. An easy way to bleed the air out is remove a heater hose or temp sender [whatever is easier] at the front of the intake. Add coolant until it is full. There will still be some air in the system but not much. Heat cycle the engine. After it cools off remove the cap and fill again at the radiator.
Saying the engine is hot is kind of subjective. Does it show hot on the gauge, is it pushing coolant out, what? If possible, buy a heat gun so an actual temperature is known. Not that I am a fan of Harbor Freight, but they are reasonable there.
There are all sorts of things that can cause overheating from a head gasket to just a radiator cap that won't hold pressure. A little more diagnostic information would help.
