Washington Square (one block from the Jailhouse)
At the literal center of colonial Newport, this park historically known as "the parade" is bound by wrought-iron fencing and features a prominent statue of Oliver Hazard Perry.  Within Washington Square is a smaller area named for President Dwight D. Eisenhower, who spent summers in Newport.  Benches, a fountain and a replica of an historic horse trough also adorn the square.  Significant buildings on the perimeter include the Colony House (where the Declaration of Independence was read aloud to citizens in 1776), Jane Pickens Theatre, Newport Opera House, and the present-day Newport County courthouse.   
Queen Anne Square (three blocks from the Jailhouse)
This sloping park situated between Thames Street and Trinity Church was created by philanthropist Doris Duke, a Newport resident who preserved a significant portion of Newport's colonial architecture.  Ironically, creating this green space (for the occasion of Queen Elizabeth's visit during Tall Ships 1976) involved razing buildings on the site.  Now the park has undergone an extensive redesign by famed landscape artist Maya Lin, whose concept was to commemorate these layers of history.  Seating areas are stone foundations evocative of structures past, and quotes from centuries-old journals have been carved into the thresholds by Newport stone carver Nicholas Benson.  The "new" Queen Anne Square reopened in May 2013.  
Perrotti Park (three blocks from the Jailhouse)
Between Long Wharf and the harbor master's office, alongside America's Cup Avenue, this comparatively modern park has a wire-frame dolphin sculpture and benches where one can enjoy the harbor, boats or a nice sunset.  Harbor shuttles and cruise ship launches arrive and depart from this location.   
Touro Park (six blocks from the Jailhouse)
Just below Bellevue Avenue, across the street from the Newport Art Museum, is Touro Park.  It contains "The Old Stone Mill," an impressive cylindrical structure of mysterious origin.  Some believe the tower was built by the Vikings; others believe it was Chinese adventurers, Scottish Templars, or Portuguese fishermen.  Still others assume the tower is a colonial-era windmill built by Benedict Arnold's great-grandfather, who owned the land in the 1600s.  Touro Park also features statues of native Newporters Commodore Perry and William Ellery Channing.   
Liberty Square (right next door)
Adjacent to the parking lot of the Jailhouse Inn is a tiny "pocket park" with two aged trees and plaque explaining that the site was a colonial mustering ground.  
Within several miles of the Jailhouse (requiring travel by car, bicycle or harbor shuttle) are additional parks and nature preserves, all well worth visiting during your visit to Newport:  Fort Adams State Park, Brenton Point State Park, Sachuest Point National Wildlife Sanctuary, Norman Bird Sanctuary.