I love Santorini and this is my 4th time visiting the island. My advice is rent a car. Even with driving being tough with the thousands of tourists packed everywhere on mopeds, motorcycles, tiny cars, it is so much better than trying to find a taxi.
Today, I spend 3 hours to get a taxi to take me 4 miles...just too hot and dangerous to walk.
I had to go store to store ans ask after I finallu wapked into a supermarket, the owner called his friend that owns a transfer service, pay cash to get a ride. No uber, taxis or any app or service anywhere.
Please share your tips if you have any, about how to find a taxi in Santorini?