For every minimal one-sided shift space X over a finite alphabet, left special elements are those points in X having at least two preimages under the shift operation. In this paper, we show that the Cuntz-Pimsner C * -algebra OX has nuclear dimension 1 when X is minimal and the number of left special elements in X is finite. This is done by describing concretely the cover of X which also recovers an exact sequence, discovered before by T. Carlsen and S. Eilers.