If you're starting out with the radiator topped off, what you're experiencing is normal. The water will seek its own level as the engine runs and warms up. As it runs and heats up it naturally expands, allowing slightly less room for water. This water will run out the overflow tube, as you described. after awhile, it reaches a point where the water level will be just above the core, making it look at first glance as if it is low on water when it is not. At this point, no more water will run out the over flow tube and neither will it overheat.
This is perfectly natural. You don't see this in modern vehicles due to the overflow tanks they have.
If it will make you feel better, drive the car w/o adding any water but keeping an eye on it to make sure it doesn't overheat. Keep a few gallons of water in the trunk just in case.
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Bill
26 Chevy Roadster
56 Buick
VCCA & AACA