There is plenty to do on Naxos, the largest of the Cyclades Islands. I was there for 8 days early October and stayed busy hiking, walking, exploring, taking the bus to various villages, eating, drinking, just a bit of beach time. I did the Delos/Mykonos cruise which took up most of a day. There is also the day cruise to Santorini but it would be a very long day, more time on the ferry than on the island. Those day cruises could be limited or shut down in Oct. depending on weather and number of passengers.
There are the Small Cyclades Islands that you can get to from Naxos which are relatively close, but ferry service may be limited for them that time of year.
it would be better to spend your time on the island, one of Greece's best all-around islands that has something for everyone.