Rangers have been known to "borrow" an exit day from before or after your requested exit day, so that, for example, a 6 person group can exit together even though there are only 5 slots open for that day. Call the permit office. They can still do things for you the old fashioned way.
And yes, as Steve says, they will not hold you to the exact day you pick. Try to do that, yes, but they realize you may be a day or two early or late depending on backpack travel on the trail.
(obviously this is not true for a dayhike). It also means that instead of trying to grab one particular day for an exit permit, you actually could use one from any of three days.
