  Let $C$ be a curve of genus $g \geq 11$ such that $g-1$ is a composite
number. Suppose $C$ is on a K3 surface whose Picard group is generated by the
curve class $[C]$. We use wall-crossing with respect to Bridgeland stability
conditions to generalise Mukai's program to this situation: we show how to
reconstruct the K3 surface containing the curve $C$ as a Fourier-Mukai
transform of a Brill-Noether locus of vector bundles on $C$.
