I went on a backpacking trip that was incredible on the Big Island.  If you have time for a longer trip you should check it out.  You start at the Waipo Valley Lookout and take the Mulawai Trail to Waimanu Valley. 

It's a rough trail but the end goal is so worth it.  There are 9 campsites in the remote valley - you'll need to make a reservation online before you go.  I did see a trail runner that did the trail in one day but you really need to spend the night to have extra time to explore. Or two...