Westport is not all beaches, waves, and fishing there are actually some really nice hiking/walking trails in town as well. One of my personal favorites can be found at Twin Harbors State Park.
The park's beach trails take off through a canopy of shore pines and soon transition to sand dunes and dune grass. While the beach is nice I love hitting these trails on overcast or stormy days. The broken light coming through the trees can really lead to an eerie and dark feeling. If you are there during the offseason you most likely will be the only one on the trails which makes it perfect for exploring and just connecting with the woods.
Twin Harbors State Park covers 225-acres and is located about 4 miles from Westport. The park has plenty of camping sites along with a cluster of cabins and yurts as well.