2011 Prius Three, 158,000 miles. This has happened twice: Car is started cold, drives normally. At less than one mile, HVAC system on Auto starts blowing cold air (AC off, outside temp in 30’s F). After a few miles, coolant temp light comes on. I shut off all HVAC, coolant temp light goes out in less than 15 seconds. Turn HVAC back to Auto, and I get heat. Thermostat stuck closed?
How do the engine and inverter coolant levels look? Do you have a scan tool capable of retrieving trouble codes and coolant temperatures? There are two engine coolant temperature sensors and several powertrain temperature sensors, and just the one temperature light on the dash, so without a scan tool, it's tough to even know which system to be thinking about. I would counsel trying to get that information first, then start guessing what the cause might be.
I noted this morning that the top of the engine coolant tank was wet with coolant. It's a fairly new tank and the cap is tight, but it looks like something is leaking. I'm going in for an oil change in a few days, so I'll have them look at it.