we only have two real elevated trails on the island. The cold spring harbor trail and the jaynes hill/mt. Misery which I recommend both if you haven’t already.
Note that the cold spring harbor trail can be very challenging especially if you're not in good shape. It's a good idea to have a trekking pole (or 2) of some sort especially on steep spots that can be slippery (either from leaves underfoot, or loose soil or wetness). And do not even think of wearing sneakers. If you take it do not overdress as you will quickly overheat. My recommendation, if you’ve never been there before, is to dress as if you were going for a long run on flat trails — except with hiking shoes/boots instead of running shoes. Think thin layers (that you can peel off) instead of a single heavier jacket or coat.
Although it's a State Park, there are no bathroom facilities although there is a path down to the CSH library that has facilities in the back or you can go around the front and go inside the library and there are facilities inside.
Especially during the winter with no leaves on the trees there or some spectacular views of the harbor when you get to some of the higher spots.
FWIW, I was in LIDL last fall and they sometimes have some odd specials (that quickly sell out) that you don't associate with a food market. I bought a pair if trekking poles for $20. Absolutely outstanding at any price, charts included on how to adjust length for your height and arm length.
Just got today's flier and they have them back in the weekly special section.
When you're done take a walk into town and stop at Sweetie Pies (maybe 1/2 mi from the park entrance) for excellent coffee and snacks. There are tables/chairs widely separated out back and away from the street.