  Mukai's program seeks to recover a K3 surface $X$ from any curve $C$ on it by
exhibiting it as a Fourier-Mukai partner to a Brill-Noether locus of vector
bundles on the curve. In the case $X$ has Picard number one and the curve $C\in
|H|$ is primitive, this was confirmed by Feyzbakhsh for $g\geq 11$ and $g\neq
12$. More recently, Feyzbakhsh has shown that certain moduli spaces of stable
bundles on $X$ are isomorphic to the Brill-Noether locus of curves in $|H|$ if
$g$ is sufficiently large. In this paper, we work with irreducible curves in a
non-primitive ample linear system $|mH|$ and prove that Mukai's program is
valid for any irreducible curve when $g\neq 2$, $mg\geq 11$ and $mg\neq 12$.
Furthermore, we introduce the destabilising regions to improve Feyzbakhsh's
analysis. We show that there are hyper-K\"ahler varieties as Brill-Noether loci
of curves in every dimension.
