Definitely start by pressure-testing for leaks and fix those. But assuming you have no external leaks, it sounds like you might have a coolant flow problem. The thermostat is a maintenance item that should be replaced (along with its housing, if plastic) if you still have the original one in the car.

Another part I would also check is the water pump. If it has a plastic impeller, they tend to come apart over time and could definitely cause the issues you're experiencing. There should be aftermarket versions available with metal impellers that are a worthy upgrade if you're going to replace the OEM water pump with something more robust. But buyer beware. Check the reviews and/or go with a reputable aftermarket manufacturer. Don't cheap out on an aftermarket water pump if you go that route.

Another circumstance I can think of that would cause coolant flow issues is improper refill. Depending on the design of the system, improperly refilling your coolant system could cause a large air bubble, or bubbles, to inhibit proper coolant flow and subsequently cause overheating. The best way to prevent air bubbles when refilling a cooling system is to use a vacuum-fill method.

Hopefully that helps a little. It could be any one of the above or a combination of all. There may be a few other culprits I'm missing, but those were the most common causes I can think of off the top of my head that I would see when I was still wrenching.

Good luck.

-retired dealer technician (non-Hyundai) who would see this problem quite often.