In Jefferson, Maine, in the late '30s, my father was asked to join a
local organization called "The Lincoln Men", or some title with
Lincoln in it. As I remember,they were like a grange, an organization
of farmers trying to survive the Depression. Any information on this
group?

I wonder if this may have been the Lincoln Brigade? It was not a
farmers' organization, but there was quite a bit of recruitment for
this group in the late 1930s:

http://www.writing.upenn.edu/~afilreis/88/abe-brigade.html

As I recall, these were mostly farmers from rural Maine. Given that
time, people and place, I doubt that they had ever heard of the
Lincoln Brigade. Thanks!

Actually, the Lincoln Brigade did solicit members in rural areas. My
father, who grew up in a tiny Oklahoma town, told me that his dad
attended several meetings in which outside recruiters tried to
interest the local residents in joining this ultra-left group.