relation between $C \cdot K_S$ and $K^2_S$ for a curve $C$ on a complete intersection surface $S$

Question

Let $S$ be a smooth complex projective surface which is a complete intersection and such that $K_S=\mathcal{O}_S(k)$, $k >0$.

Let $C$ be a smooth curve on $S$ such that $C^2 >0$.

I'm interested in a lower bound for $C \cdot K_S$ in terms of $K^2_S$.

This is obvious if $\mathcal{O}_S(C)=\mathcal{O}_S(h)$ for some $h>0$, that is, if $C$ is a complete intersection.

Nevertheless it seems reasonable to me that also for curves which are not complete intersections $C \cdot K_S$ is big when $K^2_S$ is.

Thank you