Welcome to the forum and to New Hampshire! Where are you from?
The first place that came to mind was Radeke Cabin on the Kancamagus Highway. Very Woodsy there and spectacular fall foliage but it's reserved through the foliage season already.
The Kancamagus Highway is a very popular place for leaf peepers! The entire White Mountain National Forest is also a hot spot for great foliage so I would recommend finding a hotel or cabin rental in the White Mountain National Forest region. The area offers just about everything you can imagine. From fancy resorts, inexpensive hotels to campgrounds. The problem you may encounter is many places have already been rented for the fall foliage season. The good news is that all parts of New Hampshire can easily be gotten to in a day trip. Many hotels in NH offer fall foliage packages. While you may not get that woodsy smell or feel in a hotel, this late in the season it may be your only option.
Any idea what part of NH you were planning to stay at?