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This isn't a direct answer to your question, but there are very few roads in the White Mountains that are open to the public that can't be used by passenger cars.  A Subaru Forester would certainly cover pretty much all of them.

Unlike out west, here in the northeast, there just aren't forest roads criss-crossing National Forests, and particularly the White Mountains.  Roads that get used for forestry access tend to be closed to the public, with locked gates where they meet paved roads.  The few dirt roads intended for public use are usually well enough maintained for use with ordinary passenger cars.

If you peruse some maps carefully, you can probably find a few rough dirt roads you'd be allowed to drive on, but those will be relatively rare.

There are a few more such roads in the Green Mountains of VT.  I did have to turn around in one case when I was in a Honda Civic.  I think a Subaru Forester would have done been able to continue.