  Let $S$ be a K3 surface, $C$ a smooth curve on $S$ with $\mathcal{O}_S(C)$
ample, and $A$ a base-point free $g^2_d$ on $C$ of small degree. We use
Lazarsfeld--Mukai bundles to prove that $A$ is cut out by the global sections
of a rank 1 torsion-free sheaf $\mathcal{G}$ on $S$. Furthermore, we show that
$c_1(\mathcal{G})$ with one exception is adapted to $\mathcal{O}_S(C)$ and
satisfies $\mathrm{Cliff}(c_1(\mathcal{G})_{|C})\leq\mathrm{Cliff}(A)$, thereby
confirming a conjecture posed by Donagi and Morrison. We also show that the
same methods can be used to give a simple proof of the conjecture in the
$g^1_d$ case.
  In the final section, we give an example of the mentioned exception where
$h^0(C,c_1(\mathcal{G})_{|C})$ is dependent on the curve $C$ in its linear
system, thereby failing to be adapted to $\mathcal{O}_S(C)$.
