Utilize hotels.com or other "black-out date" wholesalers and discounters.
They purchase rooms in bulk at a discounted rate and mark them up for a profit. As occupancy becomes scarce they will hold rooms back for the last minute premium. The room allocations they have will not be available to the hotel front desk or central reservations.
Hotel front desks generally have limited visibility into the "true" occupancy level anyway. Hotels allocate their inventory across multiple channels and basically lose all control and visibility over those allocations. The front desk is just another one of those channels - and not the biggest one either.
However, just because Hotels.com took 20 rooms doesn't mean all 20 of those rooms are reserved - but unfortunately to the front desk it appears as "Not Available".
Room inventories are allocated to the distribution partners months in advance. Hotels.com, Priceline and Hotwire are the incumbents - if their websites don't have availability - getting them on the phone can be worthwhile. These guys are experts - and they have more resources, influence and options than most front desks.
Additionally - there are 4 primary Global Distribution Services that get the bulk of the room inventory and they don't share with each other. This shows up on Travelocity, Expedia and Orbitz.
The easiest way to hit them all is to use Kayak.com or Trivago (or any good aggregator) so you can be certain you're hitting as many sites as possible. It does matter. The more travel sites searched - the better your odds.
Just because the front desk doesn't have rooms, doesn't necessarily mean there's no availability. It just means the hotel has used all of their own allocation.
Keep checking back every day around cancellation time - (re-run your searches) A single cancellation will trigger an available room to pop up somewhere, regardless if that site had previously shown sold-out. You should also try searching for a shorter duration (like just one night - starting on the preferred check-in date) to see if there might be rooms available some nights; but not for the entire duration. It may not be ideal but it will shed light on the true nature of their occupancy. Maybe you're searching for 7 day stay and they actually have 6 of those nights available - but you're rejected because you specified 7. If you find anything - even just one night - It's a start. Once there - you can generally stay.
Cancellation policy is typically 5pm day of arrival - but can be as long as 7 days prior to arrival if there is a big event. Call the hotel front desk and ask for the General Manager every day - (around the cancellation time - 6 or 7 pm) and also recheck Kayak or trivago. They won't call you, but they can be convinced to "pocket" a cancelled room if you convince them you'll be calling every day to check for new cancellations.
Getting on a first name basis with the GM can have huge benefits. It's not uncommon for hotels to "overbook"; knowing that inevitably some people cancel at the last minute - but still trying to fill as many beds as possible. The GM at a reputable hotel will generally help any way they can - within reason.
If all else fails; be at the hotel at least an hour before the final cancellation time on the day of (hopeful) check-in. Introduce yourself to the GM; and wait patiently in the lobby where you can spectate the check-in process.
A room will become available. The only question is - will you get it. Let us know....and good luck.