First off, there are two kinds of coolant tanks. One is a simple "overflow tank", unless you empty it yourself, it will overflow eventually. Second is actually a "coolant recovery tank". When coolant get hot and expands, it goes to the tank, and when coolant cools and contracts, water is drawn from the tank back into the radiator.
A coolant recovery tank must have a tube inside it to siphon the coolant back into the radiator, and this tube has to be immersed in coolant for it to function. The radiator cap must also seal at the bottom seat and at the top of the neck. This type cap has a gasket at the bottom for the bottom seat seal.
It is normal for coolant temperature to rise when you shut off the engine. The reason is that the block and heads are still at operating temperature, and heat transfer from the metal to the coolant can continue without any coolant flow. But there is little or no heat transfer from the radiator to the surrounding air because there is no air flow.
If you have a stock type downflow radiator with a top and bottom tank, you probably don't need an overflow or recovery tank at all if your radiator is working right and you have enough airflow across the radiator. All you need to do is run enough coolant in the top tank to cover the tops of the tubes, but leave an air gap for expansion. You may need to experiment to find the right level to fill when cold to avoid losing coolant. There is no loss of cooling capacity as long as coolant covers the ends of the tubes. There is no harm in using a coolant recovery tank with a downflow radiator, and you can fill it to the top if you have one.
A crossflow radiator is a bit different. There, if you have an air gap at the top, you may leave the top tube(s) not covered with coolant, and if that happens you lose cooling capacity. So you really need a coolant recovery tank if you have a crossflow radiator.
Actually there is a third kind of coolant tank. It's called an expansion tank, and its highest level is above the top of the radiator. The expansion tank has a pressure cap on it, and there may or may not be a pressure cap on the radiator itself. A lot of Corvettes have these from the factory, and my late model pickup has one. With the expansion tank, there is a large air gap above the normal coolant level, and the normal coolant level is above the top of the radiator. It's rare to see one on a 55-57.
