Me and my son are the lone luckiest Disney visitors ever. We had the great good fortune to of being the only tourists at the entire park one Sunday due to that horrendous string of hurricanes in 2004. We were the only people too stupid to cancel. At the last minute the hurricane veered sharply west into the Gulf. It was too sudden for people to change their travel plans.
We did not have short waits for rides, we had no waits at all. As fast as we could run through the deserted cattle chutes we just got on again, usually in the same ride car that we just got out of. For a 6 YO it was paradise.
That's a good story. I had a similar experience. I was at the Terra Cotta warriors on a trip to China in 2003. It was the height of the SARS panic. Chinese were told not to travel, and many large cities banned people who had returned from the countryside from going to work, telling them to stay home. As Foreigners, we were pretty much free to do as we pleased.
Chinese tourist attractions are usually mobbed with people. This time, 2 colleagues and I were literally the only people in the entire park, with the entire staff there to explain, give us tours, etc...
Disneyland in California does one night a year for police and firefighters. They close the park and run everyone out at about 4 pm and then at 5 start letting the firefighters and police families in. Must show fire or police ID. Open til midnight. We used to take our daughter every year. Not one line anywhere. Like you said, get off circle back through the stalls and on again. Was great.