It's not possible to give a definitive answer to your question.
As I mentioned in the other thread, the ferry routes you're looking for between the Dodecanese and the Cyclades are subsidized public service routes that operate year-round and are independent of tourist activity. Therefore, whether it's mid-October or late October makes no difference.
These routes are subject to annual tenders at the end of the year for the entire following year, and the ferry companies' published timetables depend on the route allocations resulting from these tenders. (Starting this year, these public service contracts will now have a duration of four years, instead of annual ones, this may allow for earlier publication of the 2026 schedules.)
In any case, the routes you're interested in (Kos-Astypalea-Naxos in both directions) are not daily, as they are mostly part of long ferry journeys that can take more than 12 hours to get from the departure island to the final arrival island, and the return ferry doesn't make the trip until at least the following day.
Therefore, you'll have to wait for the ferry schedules to be published to know the exact days and times of the journeys you're looking for and the number of days you can stay on an island before the departure of a ferry to the next island on your trip. Ferry schedules operate on a weekly basis; when they are published you will know which day of the week you can take a ferry between one island and another. It is also possible that you may have to depart from or arrive at an island in the middle of the night, which makes booking accommodation difficult.
