Yep - I've been looking at mytrip.net (Japanese site) and Nagoya looks solidly booked up. I guess it may be because of the national holiday on Monday (Oct 10).

With two nights in Japan, you have time to get out of Nagoya altogether. You should consider what kind of an experience would suit you best.

One previous poster suggested Gifu. That's a fair idea.

Kyoto is the "must see" city that most tourists head for at some stage - It would require you to take a bullet train from Nagoya (adds to expense, but ...)

I checked hotels in Kyoto - and they appear to be fully booked too. But you could stay in Osaka and travel from there to Kyoto to do some tourism.

If you want to get out of the cities altogether and stay at a hot spring, there are options in Gifu and Nagano.

All of the above are relatively short journeys from Nagoya.

What kind of things would you like to do during this time in Japan?