One park that I visited on a whim was Capitol Reef, in Utah. My plans for several days backcountry camping in Zion fell through because of snow fall late in season. I had to improvise. Hikers are some of the most friendly people and love sharing knowledge on the trail. One suggested I check out Capitol Reef. What a gem! It was far less crowded than Zion and absolutely gorgeous.
See the parks! Camp there! You'll need to plan 6 months in advance for permits to the most popular destinations.