It is not a matter of being allowed to enter. The airline will not allow you to board the flight in the first place, not even your US-TR flight. You must reschedule or cancel your trip.
Airline personnel do rudimentary document checks before you board, using a database called Timatic, which states directly
Passengers are not allowed to enter.
except EU ...
Take a taxi. There are plenty of taxis on Santorini. Just make sure you arrange the trip ahead of time, given the early hour. I would expect that your accommodation can help you order a taxi for the right time.