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You may have walked up to the falls at Taughannock Falls State Park. Another that is a nice state park walk is Watkins Glen. If you go to Watkins, park on the street and you won't have to pay they state to park as the village has two-hour free parking. At Watkins, if you return via the north side, there is a point that you can exit the trail at the cemetery and walk back down via the cemetery roads (needless to say, it's pretty quiet). It will bring you out to the street a couple hundred yards from the park entrance.
Maybe TC556 will stop by with some guidance as he is an avid hiker.
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Any of the local state parks are fine for day hikes.
Taughannock can be nice if there's water going over the falls, otherwise it can be a bit of a letdown
We hiked to loop around Taughannock last weekend and there's was still some decent waterflow.
If you park outside the park around Gorge Rd, etc you can skip the gate charges you get hit with in the main parking lots
https://parks.ny.gov/parks/taughannockfallsUpper RH Treman is OK, usually lots of water going over Lucifer Falls. If you decide to jump onto something less civilized at Treman you can always jump onto the Finger Lakes Trail from the Rim Trail. That'll let you connect to trails in 8 or 9 states. Too many miles of trails for a day hike.....
https://parks.ny.gov/parks/roberttreman/Buttermilk is mainly known for the falls at the lower park but its usually too packed with people in the summer for my tastes.
Upper Buttermilk park has Treman Lake which is NOT part of Treman State park. Treman Lake is pretty heavily silted in so not much to see there.
https://parks.ny.gov/parks/buttermilkfalls/details.aspxIf you want something a little closer to town you can do the Cayuga Trail
https://ithacatrails.org/site/Cayuga%20Trailwhich is close to campus . You can modify that to fit your desired time or miles. I did that trail system a couple of weeks ago.
or the co-located Lick Brook/ Thayer Preserve nature preserve which are just minutes from downtown on the west side of town near Buttermilk. Matter of fact you can hike into upper Buttermilk on the FLT spur trail from Thayer.
https://ithacatrails.org/site/Lick%20Brook%20-%20Finger%20Lakes%20Land%20Trusthttps://www.fllt.org/preserves/sweedler-and-thayer-preserves-at-lick-brook/There's a decent cidery right up the road from Thayer/ Lick Brook, if you're into that sort of thing
https://www.southhillcider.com/Too many wineries to list in and around Cayuga and Seneca lakes
Good luck with your visit
My mom grew up in Hammondsport so I spent a fair amount of time there as a kid.
You might want to consider the Keuka Outlet Trail that runs between Penn Yann and Dresden.
The Pen Yann end is more urban, the Dresden end more rural
https://www.keukaoutlettrail.org/