If your looking for some trails to walk then there are a couple of places that you can go to in Sag Harbor.
Morton Wildlife refuge has a lot of trails. One of the trails takes you down to the beach. The other ones take you in a circle that comes back to the front. You can feed the birds too. In season there are public restrooms.
Trout Pond is a fun place to go. I think there is only 1 trail but I might be wrong. I used to go there as a kid.
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South Fork Natural History Museum has trails but you should check out there website before you go.