I've probably got about 100,000 miles touring - and I do a chunk of dirt combined with hiking on my longer tours.
There are endless possibilities when you add dirt - especially in the West.
Also, some parks allow you to ride fire roads - esp. Canadian National Parks.
I've biked into Yosemite, gotten the concession company to run my bike up to Tuolumne, then hiked up the John Muir.
There's a shuttle service that will take your bike to the other side of the Grand Canyon, I've hiked it 6 times.
It's all up to you.