Chester Rod and Gun Club also has sporting clays open to the public. I believe its Wednesday nights. I went a couple of years ago, really good time and friendly people. The course is different just about every week and everybody is helpful if you are looking for tips to improve your shooting.
Google www.wheretoshoot.org and you'll find details & contact info for the region. Waldron's (Barrington NH) shoots skeet on Sunday mornings, Hampton (North Hampton NH) has a five stand on Saturday mornings, plenty going on depending where you are.