Kyushu itinerary

Hi everyone,
This is for my 10-days trip to Kyushu in Nov. May I ask for some help pls?

Day 1: Arrive via air at Kagoshima / Stay at Kagoshima
Day 2: Take morning ferry to Yakushima
Day 3: Yakushima
Day 4: Yakushima
Day 5: Ferry from Yakushima to Kagoshima / Pick up rental car / Drive to Ebino Kogen (1.5hrs)

Day 6: Short hike ard Ebino Kogen / Drive to Kumamto
[My actual destination is Takachiho Gorge but i think it will be too tiring to drive directly from Ebino Kogen hence thinking of stopping by for a night along the way. Any suggestions on where to stay? Does not have to be Kumamoto if it is a hassle to drive into the city. Any good sights along the way too?]

Day 7: Continue drive to Takachiho Gorge / Explore Takachiho Gorge / Stay nearby
Day 8: Drive to Yufuin / Stay at Yufuin
Day 9: Drive to Fukuoka and return rental car
Day 10: Fukuoka
Day 11: Home sweet home

Does this work? Actually going by the above draft, i'll have one more day. Any idea where to slot it in?

Many thks in advance!

Beppu
Asosan
Kurokawa Onsen
Yufu

Just a few place you can add that won't take you too far off you current path.

One of the first things that pops up is if you checked with the car rental agencies or not about the drop off charges - which could be very high for starting at Kagoshima and ending at Fukuoka.
For your first day, you don't say where you are arriving from or what time, but if you only have a day, do your best to see Sakurajima and the Senganen Garden. Both are the highlights of the city.
https://goo.gl/ir1cA5

If you are going to Ebino Kogen and with a car, it would be tragic if you missed seeing Sogi Falls not too far away. One of the best on the island.
https://is.gd/aIaKOW
You are not far from the Kyusendo Cave also - which is one of the best on Kyushu.
https://kyusendo.jp/
For adding a day and more places, The Mt Aso area and Kurokawa hot springs are the closest choice. Plus with a car you have the ability to see a lot more in the area - Nabegataki Falls is very nice, plus the Daikanbo Lookout when the weather is good is spectacular.
https://goo.gl/Q843Bg
The other choice, much more out of the way, but very worthwhile is Nagasaki - very historical city (much more than just the a-bombing) and nice atmosphere. The Yutoku Inari Shrine nearby is also one of the best on Kyushu.
https://goo.gl/hZVjBi
https://youtu.be/2UU_1Aip_l4

If going during the autumn leaves season, the Kuju Mountains can be very nice. Also Akizuki (east of Chikushino City) is famous for them.

Yes to everything Ken says.
You could combine circular car hires (pick and drop in same prefecture- doesn�ft necessarily have to be same town- often works out much cheaper, but do check as there are offers sometimes that waive this), with city to city trains.
And YES to Nagasaki

We did exactly the same (roughly) kind of trip as you :-)
We intended to go to a quite expensive Ryokan right after Yakushima which didn't work out as the ferry was cancelled (We got a plane ticket for the same day though). The weather of Yakushima is just something to keep in mind as it can mess up the rental car reservation and so on.

We also got a car in Kagoshima but drove up to Nagasaki instead of Fukuoka, there are actually a lot of nice places but we didn't plan too much, as we just drove by and went with the flow^^

Seems like Nagasaki is a popular choice?
What about I skip Yufuin and go to Nagasaki after Aso/Kuju Mountains area?
Drive to Kumamoto, take car ferry to Nagasaki, 2N1D in Nagasaki, then back to Fukuoka?

Will this work?

Is Nagasaki more of city or country? Will i still need a car there?

Nagasaki is a dirty industrial city. I am very surprised with your interest in environmental scenery that others are suggesting Nagasaki. It is a nice city to visit. But you don't go there for greenery.

If you are thinking of adding Nagasaki the Glover Gardens, Atomic bomb museum, Peace park Urakami cathedral are all worth the visit. They have a lovely streetcar system that takes you to most tourist sites. I personally love Nagasaki and area for my own reasons. Hope you have a fabulous time. I was in Kyushu last year and the weather was fabulous.