For those of you who will be in or passing through any one of the Hamptons this summer I have compiled a best of. 
For reaching the place I have to give you the best solution: 
The French house Mounat established in 1849, created the "Malle Bicyclette", a bicycle basket that contains porcelain plates, glasses, vacuum bottle, and knives and forks for two. Inside the basket there is also place for sandwiches. A clear picture of the bicycle is underneath. You can order it at: www.mounat.com. 
In order to fill your basket with fresh goodies you can find many picnic gourmet sandwiches and snacks at:
1. Cavaniola’s 89B Division St, Sag Harbor {+1 (631) 725 0095}: A cheese shop in the center of a big city, in Sag Harbor with knowledgeable staff.
2. Mary’s Marvelous, 209 Main Street, Amagansett {+1 (631) 267-8796}: A shop with perfect sandwiches, soups, cookies and snacks. Open 7 days a week at long last.
Hope you enjoy your picnic with good friends and family and do not forget to be prepared and organized for a perfect day out at the City's sun!!
Yours, A