Among the good folksingers I heard when I lived in Greenwich Village in the early '60s but who are not well-known except maybe to real folkie "insiders" are Josh White's daughter Beverly, Peter & Isobel Gardner (very eclectic and both with classical musical training), and Bonnie Dobson. The latter two, as well as Carolyn Hester, Guy Carawan, and Judy Henske, were folksingers I discovered early on, by way of live performances or workshops and/or records, but who never really became big names.   

Then there are people like Andy Cohen who were hardly known in the '60s and even now are not as well known as their talent deserves. Andy spent a short time in Champaign-Urbana, IL, where I met him and he would entertain me and others, playing my old Martin guitar. I don't think he had done much songwriting at the time, but his guitar work and blues singing were very impressive even back then.
Another excellent guitarist/singer/performer whom I knew in Philadelphia about 1970 was Max Yasgur's nephew Bruce Yasgur. DK if he ever did any recording but he was excellent.

Someone's already mentioned the Seattle area folks like Don Firth, Nancy Quense, Stewart Hendrickson, Reggie Miles, Mike Nelson, Bob Nelson, Stan James, and Bruce Baker.   
And in the Portland, OR, area we had some excellent folkies such as Jinx Davis, Steve Culver (the same one mentioned before as a Seattle area musician?), Bo and Paul Parker, Bob Cotta, Mary Benson (both of the group "Howling Gael"), Kate Power & Steve Einhorn, Bill Murlin. I might add Baby Gramps and Jim Page, as well as Linda Allen and Linda Waterfall.

And back about 1959, when I first "got into" folk music, I got a record by a folk trio called "The Coachmen." They were a lot like the Kingston Trio or The Limeliters, for what it's worth, but whatever you think of that type of "folk" music, they were excellent musicians.

Then there are people like our own Mary Garvey, who have been writing and singing beautiful folk songs for decades - not sure if they all were doing that in the '60s though.