Why would a 1993 Toyota Tercel overheat if there no holes in the radiator or hoses?

Flush and clean the heater core. Also check your rad cap and maybe change the thermostat

Three are other factors. One situation I've come across over the years is the elctric fan that cools the radiator starts to wear which in turn causes it to stick. This fan due to it's location takes a real beating. Splashing, dust, bugs, and everything else. With a duel fan system, it can be one or both. Start your car let it warm up and observe the fan/s operation. If not this I would look into the relay, which is the second leading cause of overheating in my experiences.

Your water pump may be failing. If it fails (pump bearings, clogs, etc)your thermostast temp goes up because coolant in the water jackets of the engine are being heated to boiling, but not being pumped through the rad to cool. Also check the tightness of your belts, if they are loose, your water pump may not work well. Have you added water without adding coolant? Try a hydrometer and check the strength of your rad fluid.

Possible that head gasket could be bad. If you've done other things and still overheating. If your engine has got very hot in the past this could be your problem.With a aluminum head and cast iorn block these head gaskets can be blown very eaisly. If you run with this condition very long head damage will eventually occur.