You can stay as many extra nights as you need (there is only a two-week limit on staying in one place). The permit you get only requires (and restricts) you to enter on the entry date shown on the permit.
If you are trying to change a DAY hike permit into an overnight, that is a different problem. There will likely be no-show permits available when you show up to pick up your permit, but the parking permits are necessary after July 10, and switching day permit to overnight may be tough.
