I'm thinking the week prior will be the sweet spot at most pharmacy sites I've looked have a very short window of appointments available (like only the current week), probably becasue they don't know how many shots they're getting that far out.
I think you are probably correct.
I have a family member who works in an Ontario pharmacy [it is part of a major chain].
They are booking appointments 2-3 days out and only taking walk-ins as a last resort at the end of the day if they have remaining doses due to cancelations that therefore need to be used [however, they have a wait list and will call people on the wait list - if a wait list person can be there within an hour, they get the dose, so walk-ins are an absolute last resort - and usually we're only talking 2 or 3 doses that need to be filled at the end of the day, if any].
They know about a week before that they will be receiving a shipment of vaccine, but not the specific date nor quantity. The day before they are told a shipment is arriving the next day, but they do not know the quantity. Day of arrival is when they find out the quantity. They do not start filling appointments until the shipment arrives and they know the quantity and can therefore plan based on the quantity.
( ETA - i.e. if they receive the shipment Monday and it is 80 doses, that is two days worth of appointments, so they will book appointments for Tues and Wed... then no more until the next shipment comes; rinse and repeat. They do not want to book for eg Thur and Fri because then they would have to cancel those if the next shipment isn't until the next week. )
Their appointments are a mix of people that received appointments through the pharmacy's online system and people that are on the online system's wait list that the pharmacy is manually calling to fill their remaining appointment slots. Right now, they are only doing people receiving first doses unless a person is in one of the prioritized groups for accelerated second dose (certain cancer patients, certain healthcare workers, etc).
There are still a lot of people on their wait list who have not received their first dose, and they also receive calls from people who have not received their first dose yet.