The commenter Oik is incorrect: Mt. Fuji is open until September 10, and the huts on the mountain are open until then as well. How do I know this? Because I summited Mt Fuji on September 9, and because that is the date the official website gives.

However, I see that you are not asking about Mt Fuji. You are asking about some other climbs you can do. (Also, you have already climbed Mt Fuji yourself. Congratulations!)

Mt Hotakadake in Kamikochi can be dangerous in October, because there is an increased risk of snow and ice. The same is true for Yarigatake nearby. If you are prepared for these risks and are experienced, these could be good hikes. Both of these summits can be done in a day or as an overnight, and both have nice huts at the top for about 10,000 yen with dinner and breakfast. Yari is a much longer approach than Hotaka. But please be aware of the risks.

Yake is a popular day hike in Kamikochi but is at a lower elevation, so you could go in and out without needing a hut.

There are some nice hikes in the Hakuba region, as well. Shirouma is a good hike there, as well as Yari (it is not the same Yari that is in Kamikochi). Shirouma can be done as a long, strenuous day hike, or you can stay at one of the two huts at the summit for about 10,000 yen. Shirouma is beautiful and traverses a famous snow field, but again, in October there is a chance of ice and snow that can make the hike dangerous—it is mostly exposed, and very steep. Keep an eye on the weather and be prepared.

Good luck!