Karpathos or Kythiria

One possibility, after your 5 week trip to Turkey, would be to go to a Greek island and then finish in Athens but it depends on where you end your trip in Turkey.
If you are close to the west coast of Turkey, you can take a ferry to Samos Island from Kusadasi or a ferry to Kos from Bodrum.

From Kos or Samos you can fly to Athens.

https://visit.samos.gr/index.php/en/

https://kos.gr/

Kythiria is in the Ionian Islands, geographically completely opposite to the Turkey-Athens axis

Hi,.....thanks for the input. I should have mentioned we will start and finish the trip in Istanbul and then fly into Athens.

Both appear to have airports, but do they have daily flights?
For less commonly mentioned places, I usually find searching for travel blogs helpful. https://www.chasingthedonkey.com/things-to-do-on-island-of-kythira-greece-guide/

I also look in European newspaper travel sections--a good resource!
https://www.theguardian.com/travel/2021/mar/12/ruins-of-dodecanese-my-life-on-greek-wild-islands-villages-greece
https://www.theguardian.com/travel/2015/sep/03/is-kythira-the-perfect-greek-island

Since there is no possibility to go by ferry from Athens to Karpathos or to Kythiria (if that is where you want to go), the only possibility is to fly from Athens with Olympic/aegean or Sky Express.

In October there will probably not be more than 2 flights per day to go to these islands.

Tourist infrastructure is probably more developed in Kharpatos and from Kharpatos you can easily reach Rhodes by ferry.

Anyway, on one island or the other, in October you will not be disturbed by the crowd.

I spent a week in Karpathos and enjoyed Olympos a lot. It’s a quiet island but I was a little bored. We had some great food. I did not make the arrangements so I can’t say exactly where we stayed other than it was not far from the airport. The house was beautiful and the view was stunning. You will need a car.
https://www.greeka.com/dodecanese/karpathos/

I have not been to either in mid-October but I would have to thing there will be a whole lot more to do on Kythera at that time of year than on Karpathos. Actually, any time of year.
Given this choice I would say Kythera but by mid-October the "quiet" islands are really, really quiet, as in far less availability and options.
One idea might be to rent a car in Athens and arrange to return it to the airport when you leave. Getting to and from the airport in Athens is a bit of a hassle and takes about an hour, but the car rental office is directly outside the terminal which makes for convenient drop-off and departure.
If this idea appeals to you, the best island to drive to is Evia, as there is a bridge, and it's large enough to have plenty of options.