"I've never been able to get my permit on any day other than the day my confirmation letter says I will be entering the park.  As far as I know and in my experience, I've never heard of an exception being made to the daily quota."

I've had different experience, but using the Atikokan office may give me more opitons.

I have been able to change my entry date to one day earlier than my reserved date, at least twice - that depends on availablility of entry permits for that earlier date of course.  Typically, I plan my long drive with a "cushion" in case I can't make good time traveling.  So I often get to Atikokan the day before my reserved entry - if its early enough when I get there, I go ahead and try to rearrange my entry date - I either add a day to my canoe trip, or shift my timing a day ahead.

I have also picked up my reserved entry permit the day before my entry date on several occasions - you typically can't do that until later in the afternoon though - they don't want to hold up the people entering on that date - so it depends on time of day and how busy the office is, but I've been able to pick up the permit the day before I enter the park.

That'susing Atikokan, I don't know how the remote ranger stations like Beaverhouse would handle it, but I'd think it would be the same?