Looking back over this thread, two people have mentioned Mt. Kahtadin in ME. I think it's kind of telling, because Mt. Kahtadin isn't a National Park. It's a State Park. The National Parks are great, but there are so many levels of parks in this country. Counting private property that's open to the public, there are even more places to go.
One of my favorite places to hike is
Mt. Tumbledown in ME.
It's a short day hike. Takes 3-4 hours to get to the top. Once you are there, you have a spectacular view, a mountain lake to swim in, and all the wild blueberries you can eat. And it's owned by a lumber company. If it was a National Park, there would probably be a parking lot on the summit, like Cadillac Mt. in Acadia National Park.