Let $X$ be a K3 surface. Let $L$ be an ample line bundle on $X$. When/how can we say that any smooth curve $C\in |L|$ has maximal gonality $k=[\frac{g+3}{2}]$ and Clifford dimension 1. Is there some criterion to check if the ample linear system satisfies that? Thanks!

$\textbf{Edit:}$ I am interested in the specific case when $X$ is the Kummer surface associated to an abelian surface $A$.