I went through something similar a couple of years ago. In my case, I told them I was in a real rush, which I was because I was in the region for work related stuff and had a flight to catch in a couple of days, and they somehow found a slot for me , if I recall correctly, in 3 or 4 days. They used to interview only on certain days of the week. Not sure if things have changed because I heard they are processing quite a few H1s these days but it might be worth a try to ask once you get there. The guard at the bank, who takes in the papers, may not be the friendliest guy, but its worth a shot.