As a general rule of thumb, most US highways will not be problematic, and many numbered state highways also are fine. US4 to NH11 to US3 looks to me like it should pose no clearance problems.
In general, one doesn't find low overpasses (or very many overpasses of any sort) in rural areas; one may occasionally find bridges with low overhead clearance, either covered bridges or iron bridges. One may also find tree branches that are not kept as well trimmed as maybe they ought to be, though not so much along the more major roads. (Even in rural New England people rely on tall trucks to deliver and transport stuff, and those trucks go on the same roads as you and I.)
