Hanover, PA? I will defer to Hotoven.
Hanover, VA I know nothing about, my guess is you'd need to head west about 100 miles to reach some hills.
Hanover, NH is within striking range of both the Greens and the Whites. Try Mt Moosilauke at the southwest end of the Whites to get above treeline. Further south, Mt Cardigan is close to Hanover and offers tons of bang for the buck, and you might also consider Mt Kearsarge (a.k.a. Kearsarge South). A couple of small peaks can also be reached by following the AT starting right in Hanover.