Sorry if this is way too late to be useful, but the Greenbelt in Staten Island is probably your best bet at a great day of hiking and a fun day trip in the NYC area. If you've never been to SI, hear me out. Visitors from elsewhere in the city:
-Start their day with a half hour boat ride, with the Statue of Liberty on one side and the sun rising over Brooklyn/the Pacific Ocean on the other.
-have access to over 30 miles of trails, which you can access almost as soon as you step off the boat! (it's possible to walk through/alongside parkland for the ~4 or so miles from the boat to the Greenbelt)
-can visit forests, meadows, swamps, beaches, and the tallest natural points on the eastern seaboard - all within a 12 mile hike!
-end their day in the borough of NYC with the most small restaurants. Thai, Mexican, Chinese, Sri Lankan, North African, Italian, whatever you want, you can find it on SI nearby a green space or on your way to/from the ferry!
I'm more than willing to talk about specifics, let me know if you're still reading these replies and if you are considering a trip to our fine Island.