If you can't snag a site along Gale River Rd or you are looking to hike the Twin Mountains, then continue a little farther north on US 3 near Twin Mountain (a little north of the Beaver Brook Picnic Area) and try your luck on Haystack Rd. This road is closed in winter and will open in late spring after the road has dried out and any repairs have been made. In 2019, it opened in mid-June. Check the the White Mountain National Forest FB page or website (FB is usually more up-to-date) for road status, including closures in the autumn.
When open, you can drive in to one of 11 numbered, wooded sites and camp up to 14 nights for free. You may have up to 3 vehicles and 15 people on a site. The only amenity is a fire pit, but the road ends at the Little River, offering fishing and swimming options. The parking lot at the end of Haystack Rd serves as the Twin Mountain trail head. You may see moose or bears in the area.
Franconia Notch State Park is just south of here, with its hikes, recreation trail, echo lake, and the Flume. Head east on 302 and you'll find yourself in Crawford Notch State Park with waterfalls and hikes.
Forgot something? Fosters Corners just a few miles north has an eclectic selection of essentials as well as gas. Or take I-93 north to Littleton to a wider variety of shops.