Split-Hvar-Korcula-Dubrovnik-Cavtat or vice versa October?

With the caveat that I have not been in early October, I find the season to be extending in many places and if the weather is fine, I would want to stay on an island.
I would look carefully at the travel times on the Krilo and jadrolinija websites (there are a couple other providers also) first, and I would also make a personal “must” list of your mainland attractions, then see what’s workable with your remaining time.
I’d also be quite comfortable playing it by ear at that time of year.
November and late October could be iffy, but at the beginning of the month, I would feel completely secure planning a couple days on an island. The alternative is six days each in Dubrovnik and Split, and I’d rather be in the tiny villages—your mileage may vary as we used to say!

In terms of schedule you are early enough that it will be fine. It is shoulder season and things will be starting to wind down but there doesn't look like there are too many restrictions on ferries in early October this year. In general ferry schedules look like they all change in mid October or at the end of October this year. It is not as if everything shuts down on 30 September - most places will start closing over the month of October or November. The islands will be starting to get a lot quieter and they may not be so exciting but it seems like this is why you would want to visit.

As for day trips that will be a little bit harder, it is difficult to see more than one island in a day because there are very few boats that go between islands. The only real way to do multiple islands in one day is to use a private boat.

Weather will be cooler and more random than in the summer but it could also be a nice time to visit. It is still early enough that I would not be too concerned about weather impacting ferries. Of course this is possible but it is basically same risk as would be usual.

If you were interested in islands I would start north and work south taking one of the daily inter island ferries (Krilo have one, TP lines have one, not sure about other Krilo) and being flexible in case it is not running due to weather, which is unlikely.

Thanks so much, very reassuring. We'd also like to make a day trip to Mali Stom for oysters but unsure where to fit it in and how to get there. Would you suggest going from Korcula? It doesn't look like the ferry goes near the town. Private boat? Or should we take a bus or hire private driver from Dubrovnik?

You should be able to take a bus from Dubrovnik, spend a few hours, then continue on to Korcula by bus/ferry (the bus takes a ferry), or grab a catamaran at Orebic. If that’s too onerous when you look at schedules, then I’d book a driver or just rent a car.

https://autobusni-kolodvor-dubrovnik.com/en

Also this page has some info about oyster tasting since you’ll need to suds out exact location you want to go, Ston vs Mali Ston, etc.