Many small hotels (self-described as minshuku or ryokan) are not bookable through the Internet. Such hotels often have availability even long after all of the Internet-bookable options have sold out.
Booking is made by phone. Obviously, this require knowing some Japanese, as very few have English-speaking staff.
The main difficulty is to find these small hotels that often have no Internet presence. Here is the best way I have found:
旅行宿泊
(tourism accommodation)地図検索への切替
to switch to map modeAs pointed out by Mazyod, another option is AirBnB, they might have different occupation patterns than usual forms of accommodation, because some people rent their place precisely while they go on vacation.
One other solution can be urban camping, which is tricky but feasible in Japan with a bit of attention. If travelling with a low-cost airline, remember that tents might not be allowed as cabine luggage.
A last resort is cited by Wikivoyage:
24-hour comic book library/internet cafes known as manga kisa, are common […]. This is one of the cheapest ways to crash […]. No bed, but you have a comfy chair and a PC and/or DVDs if you can’t sleep. Later in the evening, karaoke boxes often offer discounted prices for the whole night, they usually have a couch you can sleep on. Most of these cyber cafes charge ¥1500–2500 for 8 hours.
Credit:stackoverflow.com‘