For \(N \ge 4\) we classify the \((N-3)\) -degenerate smooth CR maps of the three-dimensional unit sphere into the \((2N-1)\) -dimensional unit sphere. Each of these maps has image being contained in a five-dimensional complex-linear space and is of degree at most two, or equivalent to one of the four maps into the five-dimensional sphere classified by Faran. As a byproduct of our classification we obtain new examples of rational maps of degree three which are \((N-3)\) -degenerate only along a proper real subvariety and are not equivalent to polynomial maps. In particular, by changing the base point, it is possible to construct new families of nondegenerate maps.
