The Zariski-Fujita theorem says that on a projective variety $X$, if a Cartier divisor $D$ is ample on its base locus, then some positive multiple $mD$ is base point free. I'm wondering if the following related statement is true.

In the same setting, suppose the base locus $Bs(|D|)= \bigcup_{i=1}^{N} C_i$ is a union of curves, and $D \cdot C_i >0$ for $i=1, \ldots , k$, i.e. $D$ is ample on the first $k$ curves. Then is it true that for some $m>0$, $C_i \nsubseteq Bs(|mD|)$ for $i=1, \ldots , k$? Specifically, that $Bs(|mD|) \subseteq \bigcup_{i=k+1}^N C_i$ ?

I'm interested in the case where $\dim X=3$ and $X$ is smooth, but I don't think that's particularly relevant. Any help is appreciated.