Just down the road from the Bridges Inn, you’ll find a nice trail called the Ashuelot River Rail Trail  It runs alongside the river, quite scenically, from Keene to Hinsdale, on the Vermont border.
Occupying the route of the former railroad, the tracks have been removed but the trail still follows the same path through the woods, passing farmhouses.
The trip from Swanzey to Keene is quite lovely.  It is about six miles and will bring you by the Cresson Covered Bridge.  There are also great views of Mount Monadnock.  The trail is 23 miles long, most of which is south of the Bridges Inn, in Winchester and Hinsdale.  Skipping the Keene portion, bikers leaving the Inn could travel 34 miles (17 miles each way), to the Vermont border and back.
Another popular bike route from the Inn is the Covered Bridge loop.  Clicking the previous link brings you to a map with two connected loops, for a total of twenty six miles.  It is possible to do other loops, though, of various duration.  One such route would bring you to Ashuelot to see the most picturesque of the local covered bridges, the Ashuelot bridge.