Unitarians broke away from the Congregational church (New England Puritans) -- or the Trinitarian Congregationalists broke away from the Unitarian Congregationalist, I guess, depending on how you look at it! I was surprised to see googling that many Unitarian churches keep the term Congregational in their name.
Those for whom Unitarianism (or Unitarian-Universalism) didn't go far enough morphed into Transcendantalists. According to Norman Pettit (a scholar of American Puritanism and its descendants), Ralph Waldo Emerson left the Unitarian Church shortly before he would have been thrown out.
Long before there were Congregationalists there were Unitarians.