If you can rent a car for at least part of your time in Kyushu, your options vastly improve and you can see a whole world of great sights.
Mt. Aso is one example. The area is much more than the crater itself (and at times it is closed due to poison gas emissions) and there is a score of great other places - but driving is the only practical way to see them - Nabegataki Falls, Yamabuki Springhead, Daikanbo Lookout, etc.https://goo.gl/RwpnU8
Some others are doable, but much more troublesome relying on slow trains and buses. In Saga, there is the Arita Porcelain Park and Yutoku Inari Shrine - two absolute gems
, and really with your amount of time should not be missed. But while the shrine is reachable by train and bus with a moderate level of waiting, the Park really needs a rental car or taxi for at least part of the way.https://goo.gl/dnqT25
If you'd like a cheaper place for long term rentals, you could try https://www.chintai-jidousha.jp
and they have some cars for as low as 50,000 yen for a whole month. As I recall though, they also had some rates for 1 or 2 weeks as well. You also might look at getting a Kyushu Expressway Pass.http://global.w-nexco.co.jp/en/kep/
You should know for your planning that while they say 'expressway', in a lot of areas the speed limit is a meager 80 kph. Some expressway sections are also crammed full of radar-photo speed traps. Since they're immobile, some car navi systems actually tell you when you're approaching one.
Your timing is also perfect for seeing the autumn colors - at higher elevations, you can see some in October, and from mid-November, you can enjoy most of the rest.https://www.kyuhoshi.com/japan-autumn-foliage-forecast/http://allabout-japan.com/en/article/478/http://www.rurubu.com/season/autumn/koyo/
(look under Kyushu - ãB)https://koyo.walkerplus.com/https://tenki.jp/kouyou/ranking/9/https://www.jalan.net/theme/koyo/ranking/kyushu/
Some famous places include Yabakei, the Kuju Mountains, Mifuneyama Rakuen (Saga), Daikozenji (Saga), Kirishima (Kagoshima), Akizuki (Fukuoka), Kawachi Fujien (Fukuoka), Raizan Sennyoji Temple (Fukuoka; get there before 10 AM or no parking left), Nomiyama Kannonji (Fukuoka), and countless more.
Fukuoka is not Kyoto of course but there are many good sights to see - it's just that most don't know them and then start spreading misinformation like there's only good food there or it's just a gateway city. If you drive, the Shimin no Mori is another good autumn colors spot, and the Nanzoin Temple is one of the finest in Japan - yet virtually unknown to even the well traveled.http://www.thejapanfaq.com/fukpic.htmlhttps://goo.gl/X3LDaE
The true jewel is Nagasaki though. The city is truly unique in Japan, plus as mentioned you could see Unzen, Hirado, Huis Ten Bosch, plus Sasebo, the Bio-Park, and other good places.https://goo.gl/hZVjBi
And since you mentioned hiking, you might look through this site:https://japanhike.wordpress.com
For a whole range of places in Kyushu, you could also browse here:https://goo.gl/Q843Bg