2.5 days on a throu hike of the AT in Massachusetts, the northern section. It was in January, but still I would have thought that I would have seen someone in the backcountry. Road crossings, and walking roads and through towns, and my 1 town stop where the exceptions along with residents looking out windows from houses close to the trails in some places. So after all that I guess I was not alone. I was the only one on the "trail" that I could see. I don't remember exactly where I was when there where no more people in site "on the trail", but I do know that the first person I saw after approx. 2.5 days was just short of the top of Mt. Greylock. Of all people to see after that long it was a (not sure if this is still PC) deaf person. He signed in layman’s terms “almost there”.
There was another time I was alone hiking the hole way up to HoJo’s. At the time I was working for American Money Collectors, oops I mean the AMC and I got off of work at something like 8pm after dish dogging. I decided to hike up in the dark (and yes I radio’d up to make sure they new I was coming in the dark winter weather). Obviously I would see no one hiking at 8pm at night in winter on the Tuck’s Trail, but for some reason, maybe because it is normally a very heavy use trail it struck me as kinda scary. I kept hearing “stuff” in the woods. I was all spooked out.
Try bushwhacking some around. I’m sure if you research some you probably could do several days in the Whites totally off trail and possibly see no one. You most likely would not be able to hit any peaks, but still a cool idea. Have you thought about kayaking. Being a river in the middle of no-where is a good place not to see anyone for a while!