Mechaniker: if the starting state is Maine, the next state must be New Hampshire, because it’s the only one that borders Maine. After that you can only go to Vermont or Massachusetts.
If you go to Massachusetts first, you then have to hit either New York, Rhode Island or Connecticut next. If you do New York, you’re stuck because you’ve cut off Vermont on one side from R.I. and Conn. on the other, so stay out of New York just yet. If you go from Mass. to Conn., same problem: you have to either miss Vermont or R.I. after you’ve taken care of New York. And if you do Rhode Island now, your only following choice is Conn. which will again lock you out of Vermont after you then move on to the only option for the step after. So going from New Hampshire to Massachusetts is out. You have to do Vermont after New Hampshire.
So after Vermont, you have the same problem if you next try to do New York: three New England states and the rest of the country, and you’re unable to get to one group to the other. Which means after Vermont must come Massachusetts. If you go to Connecticut after that, Rhode Island is cut off from the remaining states, so that doesn’t work; you have to do Rhode Island, then Connecticut. And at this point the only next step is New York.
(Notice that New York and New Hampshire are the only states that make the map of the Lower 48 fall into two unconnected pieces if you remove them. That makes them special in a problem like this.)
So, to recap, first seven states must be Maine > New Hampshire > Vermont > Massachusetts > Rhode Island > Connecticut > New York, and they must be in that order. After that you do Pennsylvania and New Jersey, and it doesn’t really matter which one you do first, or if you do them one after the other right now. Other places along the way where you’re going to have your route tightly constrained for you are the Washington/Oregon/Idaho set and a couple of overlapping triplets (the Carolinas and Georgia, and Georgia/Alabama/Florida).