I would have to agree the Franconia Ridge is a stellar hike.
However, if I were going to go out and hike for a week in the Whites, I would definitely hike from Crawford Notch to Gorham. This of course is easiest done by following the AT the whole way. Total miles one way is about 55 miles I believe. Difficult terrain, but totally doable.
As far as camping goes - unless you want to spend an arm and a leg - STAY AWAY FROM THE AMC HUTS. They’re expensive. I second Big B. Look into some of the RMC huts, ie: Crag Camp. You will have to travel a few tenths of a mile off of the AT to reach these places, but at 12$ a night average - it’s worth it.
I do know that there is a pay campsite at Mizpah Hut, and there are also some tent sites north of Mt. Madison. North of Pitkham notch, you can rely on Shelters for staying the night.
This section of trail is just one of my favorites for sure. Good challenging climbs, amazing views, and an opportunity to resupply if need be.
Hitching would be pretty easy to do also - should you only have one car.
Enjoy!
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