Many of our Northern states will be chilly in Nov. Plan accordingly. I suggest you try couchsurfing. Free bed/couch in many of our cities/towns. Hard to beat a free room! One problem is getting to them w/o a car. Our public transport system is not as handy as European countries. Renting a vehicle for months can really put dent in your budget. Crunch some numbers on renting a vehicle vs bus, train, plane from city to city. Amtrak has multi day passes, see if that might work for you. Compare with buses. There are others as well.
The only way I can see your spontaneous: "just do something as it comes along" is to drive. Either a rental or your own vehicle. With trains, buses, planes you need to make reservations. Buying/selling a vehicle can make sense if you spend more than 6 weeks in the US. I can help if that is something you might want to consider.
Good luck with your planning.