As in Europe, in the USA it does depend on your mode of transport, your willingness to find alternative travel options, the actual place you will be at any given moment and how flexible you are in what kind of accommodation you are accepting.
If you travel by car, book online a few days ahead of time and are flexible in your length of travel day, and are willing to accept a wide range of accommodation, you do not need to worry at all.
On the other hand, if you travel by train or long distance bus and you will only have one or at most two options for travel per day and not a lot of flexibility, and you are picky on the kind of hotel you stay you better make sure you have a kind of lodgings you are happy with before you are stuck in a town with no alternative and no transport out.
Most people will be somewhere in the middle. But the general rule is, when you have more options to get to a hotel somewhere else, you need to worry less to book ahead.
Outside of local holidays and popular weekends, with a car you hardly ever even need to book ahead, as long as you are willing and able to stretch your day by a bit and have checked out there are enough hotels/hostels/campsites with cabins or whatever you find acceptable within reach.
If you travel by 'once a day' transport, like trains in the USA, you better make sure a hotel (or whatever) holds space for your arrival, but in many cases booking a few days ahead will do.