  Let $C$ be a curve on a K3 surface $X$ with Picard group $\mathbb{Z}.[C]$.
Mukai's program seeks to recover $X$ from $C$ by exhibiting it as a
Fourier-Mukai partner to a Brill-Noether locus of vector bundles on $C$. We use
wall-crossing in the space of Bridgeland stability conditions to prove this for
genus $\ge14$. This paper deals with the case $g-1$ prime left over from Paper
I.
