The absolute best time to wonder around Oia is early in the morning. And by early I mean wake up for the sunrise. I know vacations are not meant to wake up early but do yourself this BIG favor and wake up.
The lighting is phenomenal, the breeze is great and you have the whole place to yourself.
Another plus? The intoxicating smell coming from the bakeries. You pass by them and all you smell is the fresh bread and Greek donuts. It beats the smell of sweaty and overly perfumed bodies from during the day, I can tell you that.