Did you try Google? I just did and came up with the Nooseneck Hill Archery Club, apparently still in operation. That's a place to start your inquiries.
http://www.bizapedia.com/ri/NOOSENECK-HILL-ARCHERY-CLUB.htmlAlso found a guy selling a Nooseneck Hill Club member button from 1934:
http://www.worthpoint.com/worthopedia/pinback-button-nooseneck-hill-club-292994442Looks like there are bunch of fitness clubs, "gentlemen's" clubs, golf courses, gyms, etc. on Nooseneck Hill Road in and near West Greenwich and Exeter RI. Most have "Nooseneck" and "Club" in their name.
You might also try calling the Narragansett Gun Club at 45 Nooseneck Hill Rd, West Greenwich, RI 02817.
Took about seven minutes on Google to come up with that info. Just gotta chase the leads.