Thanks TrailBeast!
I've got one of those hose nozzles with the graduated nozzle for different size hoses. I probably haven't use it in 30 years. It also has a port for an air hose. I was planning on using it on the heater core but will take you suggestion and also use it through the empty thermostat housing. Based on your suggestion, I'm assuming I want to keep the gunk from the block out of the radiator and heater core.