Thank you very much for the answer! I am not sure how old the cooling system is so a total replacement seems in order. Dad also would be grateful for the advice (except he could have done it himself). The Special was his baby (except for mom of course) but I keep it alive in his honor: RIP. The 1957 Buick Special Riviera 46-R was my parent's first new car; here is background on the era and detailed specifics about the car: https://blogsmithconsulting.blogspot.com/2017/07/the-buiee-requiscent-in-pace-leo-j.html
That is an outstanding car! Love old Buicks and especially anything with a nailhead. I agree with a complete rebuild of the cooling system along with a flushing of the engine block and heater core also. Sometimes a good flushing is all that is needed, along with a new, properly rated and proven radiator cap. I've seen a number of cars that were overheating because a radiator cap spring was going bad or someone had replaced a correct cap with a cap of much lower rated pressure. It's really important to have the correct pressure cap on a cooling system, for it to cool properly. Most auto stores have a radiator pressure tester, that can also double as a cap tester, and will do that testing for you. Nearly every Auto shop would also have those tools also and would be happy to help someone with such a cool car test a radiator cap and or radiator.