Melalvei, you may not find this relevant since I've never traveled in the US, but here are some of my experiences of travelling alone as a woman, hiking and camping in a tent.
Norway: in the woods and in mountain areas, I feel safer than in my own bed. Never met anyone frightening in any way.
France and Spain: rural and mountain areas, people were almost all extremely friendly, anxious to help, surprised I was alone. One strange encounter with a lonely shepherd who suggested I join him in his hut for a romantic encounter, but when I politely declined he just sighed and shook his head. I'm not making this up. A bit more staring and catcalling in the cities than I liked, but nothing scary.
Romania and Bulgaria: rural and mountain areas, plenty of strange looks, but no unpleasant situations. I was pretty much left alone as an obvious outsider (blonde, wearing western gear). One creepy guy in a village campground who hid in a hedge and started making kissy noises at me, but when I marched up to his plot later and yelled at him in front of his wife he completely ignored me.
In general my experience is that people are nice. And a lot more worried about you than you are. You stick out as a lone woman, and may find it hard to just blend in. Confidence and a lowkey 'boring' manner helps.
Winter riding is much less about badassery and much more about bundle-uppery. - malkin
1995 Kona Cinder Cone commuterFrankenbike/Selle Italia SLR Lady Gel Flow
2008 white Nakamura Summit Custom mtb/Terry Falcon X
2000 Schwinn Fastback Comp road bike/Specialized Jett