5 Days between Yakushima-Beppu

I have 5 days after checking out of my place in Yakushima before I go to a hostel in Beppu. Any suggestions on where I could go during this time? Prior to Yakushima I will have spent 2 days in Fukuoka, and after Beppu I will be heading north towards Narashiki and Tokyo.

Appreciate any advice on where to go spend time between Yakushim and Beppu. Thanks.

1) Beppu
2) Yufuin
3) Fukuoka
4) Dazaifu
5) Mount Aso
1+2: you can buy yufuin 1 day rail and bus pass = 2270 yen,valid JR and kamenoi bus
3+4:Fukuoka tenjin to dazaifu 1 day pass by nishitetsu bus =820 yen
5 :minami aso railway 1 day pass =1200 yen

Time of year?
Car or public transport only?

Ferry or flight from Yakushima to Kagoshima or Fukuoka?

Lots of options, and the depend on the above answers.

Kagoshima/train:-
Kagoshima 1d
Kumamoto 1d
Yanagawa 1d
Hita 1d
Yufuin 1d

Kagoshima/car:-
Kagoshima 1d
Kumamoto 1d
Aso 1d
Takachiko 1d
Yufuin 1d

If warner months, the going via Miyasaki instead might be nice. Seeing Aoshima, Nichinan coastline sights etc etc.

You can see the rest of Kyushu by spending about one day each prefectures of Kagoshima, Miyazaki, Kumamoto, Nagasaki, and Saga. There are 7 prefectures in Kyushu, and Fukuoka and Oita (Beppu) are already in your itinerary.
https://www.japan-guide.com/list/e1108.html

Consider Kyushu Rail Pass. https://www.japan-guide.com/e/e2361_05.html

Here is my current schedule:

9/24,25 : Fukuoka Accomodations booked
9/26: Figure out how to get Yakushima
9/26-29: Yakushima Accomodations booked

<GAP>

10/5-7: Beppu Accomodations booked
10/8-10: Kurashiki Accomodations booked
10/13: Fly out of Naritata

That's the plan for now.

The other option is:

9/24,25 : Fukuoka Accomodations booked
9/26: Figure out how to get Yakushima
9/26-29: Yakushima Accomodations booked
9/30: Kagoshima
10/1-3: Beppu
10/4-6: Kurashiki
<GAP>
10/13: Fly out of Narita

Note any accomodations listed as "booked" are cancelable, but rather not cancel as they are pretty decent lodgings for the price.

Flight from FUK-KUM is 24,000 Yen. Train ticket from Hakata to Kagoshima is 10,000 Yen + Toppy Ferry to Yakushima 12,000 Yen. Seems easier to fly at this point? I plan to get a 14 day JR pass and have it activated October 1, inline with my October 14 departure. I won't be eligible to drive and traveling solo.

Thanks.

Ooops, should clarify these are the NIGHTS booked, and all bookings are fully refundable at this point.

As already noted, flying is convenient, but the airfares are quite high, and not eligible for any special fares for foreign tourists.
There is certainly no shortage of good places on Kyushu though - but if seeing Yakushima, spending some time in Kagoshima is also a good idea. Sakurajima is quite a unique place (as is Aso), bu there is more - the Senganen Garden is very nice, as is Amami no Sato. And the shochu is something worth trying.
Takachiho makes another good place to visit. If you want a real treat and the most interesting city on the island, I'd say go see Nagasaki. Unzen is another great place close by, but better if you are renting a car.
Saga also has a number of good sights, and mostly ignored by the majority of tourists.
https://goo.gl/Q843Bg