2005 Hyundai Tiburon heater

With the engine at full operating temp, locate the 2 heater hoses where they go into the firewall and feel them to see if they are both hot to the touch. If they are both hot, the heat in that vehicle is controlled by a blend door that regulates heated air flow. it is operated by an electric motor/actuator. The problem can be that the actuator is stripped or inoperative or the door itself could be damaged. This is what needs to be determined by examining the actuator and see if it is responding to heat change commands or not.

you said on a really cold day it only gets warm. that tells me your engine is running to cool. watch your temp gauge if you have one. does it sit lower than normal on those really cold days? if so, you probably need a thermostat . they keep and engine fro running too cool as well as too hot

We don't know that the engine is running cool yet. Please follow the instructions I gave you and that will take us in the right direction.