I agree with
Tizzette about bringing a lightweight, medium sized, cross body purse that can be flattened in your carry on luggage.
I hope my story sets your mind at ease: I'm someone who loves purses and when I'm home, I swap them out on a day-to-day basis...and they aren't particularly small.
Yet when I'm traveling, I somehow manage beautifully with a one much smaller REI travel purse I bought years ago. I have been able to fit anything I need for a day of touring inside (tissues, small camera, pens, small notebook, iPhone, a few band aids, lozenges, sunglasses) inside.
travelmad478 was spot on about the temptation to fill up a day pack; even if I wasn't the culprit, the SalishSea family regarded me as their porter when I used a daypack instead of a purse, and I was lugging around
their water bottles and jackets during the day.
I was in pain by the third day. Never again.
I envision you traveling nice and streamlined through Greece