Pencils and nets of small degree on curves on smooth, projective surfaces of Picard rank 1 and very ample generator

Let S be a smooth, projective surface of Picard rank 1 and very ample generator embedding S into P^n. Let C be a smooth curve in O(m) for m \geq 5. We prove that any base-point free, complete g^r_d on C for r\in{1,2} and d small enough is cut out by a hyperplane section restricted to a multisecant (n-r-1)-plane.