The Whaling Museum is open year round as are their historic sites. however, their walking tours end with Columbus Day Weekend.
Some of the local transportation is available year round. So you could take the WAVE or bike to Sconset and other places on the island. If you walk downtown on your own you may benefit from a visit to the Whaling Museum prior, they show a film and explain all the sites and their history you may pass on your walks.
Brant Point Lighthouse is a nice little scenic walk.
Not all restaurants stay open after Columbus Day and some of those open operate reduces hours.
Personally I would enjoy the peace and quiet of off season together with the more reasonable rates that come after Columbus Day.