A stock Supra in good condition does not overheat. They were literally Toyota's top of the line car and are a wonderful vehicle for extended driving. A coolant system check, fluid replacement (oils and coolant), bad head gasket test and reasonable expectations are recommended.
On MKIII Supras, overheating is very often a symptom of a bad head gasket (BHG). Stay away from any Supra with any indicaton of current overheating problems unless the price is greatly discounted. If you have done research you should know that the MKIII Supra is particularly susceptible to BHG. Nothing will stop a BHG car from overheating except for frequent replenishment of coolant as a short-term measure in some cases and a proper head gasket replacement that should go beyond Toyota service manual recommendations. There is much available information on this and the gist is a higher tensile strength bolt (ARP head bolts or studs) and a higher head bolt torque is required. A metal head gasket (MHG) or a composite head gasket may be used, The primary factors in this choice is head and block surface preparation, which needs to be very smooth for a MHG, and the level of performance upgrades you may want to pursue. If you are not planning to heavily modify the car, a composite gasket and propoer cleaning of the deck and milling of the head should work fine if used with the ARP bolts and higher torque.
I would suggest that you test for a bad headgasket prior ro buying the car since you are looking to make a long drive immediately after purchase. You may want to specify this test as a requirement for vehicle inspection if you plan on hiring a professional inspector to check out the car. You can also buy a test kit at most any automotive store and do this yourself.
A modified Supra may be more of a challenge to size up. Turbo cars, especially highly modified ones, tend to produce more heat unless the cooling system is also upgraded.
Again, the car must be in good shape and tested for head gasket / cooling system issues given that you want to make a long drive and probably want it to be enjoyable and trouble free. Good cars typically have had less owners and the last owner has had it for a while and is replacing it for reasons that do not indicate a problem with the car. Ideally, the car you want to buy would be a one owner car that looks like new and is owned by someone who obviously has taken care of it. These are becoming very hard to find but are worth the wait and additional cost. You should expect to pay at least $6,000for a nice MKIII without issues and a reasonably well documented history. If you locate a car, in the Tampa area of Florida, I would be willing to provide a cursory check for you. I am sure other members of this forum would also offer you the same courtesy.
Have reasonable expectations, make good diligence and have good luck.