  Given a non-hyperelliptic curve $C\in\mathscr{M}_g$ and $2\leq n\leq g-2$, we
prove that the generic fiber of the Gauss map on $W_n$ has one element and we
characterize its multiple locus. Assuming that $C$ doesn't have a
$\mathfrak{g}_{n+k+1}^{k+1}$, for $1\leq k\leq n-1\leq g-3$, we solve the
problem of reconstructing each $\mathfrak{g}_{n+k}^k$ and the dual hypersurface
of the image of its associated morphism, through information encoded in the
Gauss map. For this purpose we introduce the notion of
$\left(n+k\right)$-intersection loci and we study their dimensions. In the
hyperelliptic case we prove that the image of the Gauss map is a union of sets
whose closures are birational to their complete $\mathfrak{g}_{n+k}^k$, for
each $1\leq k\leq n\leq g-1$, and that these also contain a copy of the dual
hypersurface of the image of its associated morphism. From the case $k=n$ we
deduce that the closure of the image of the Gauss map is birational to
$\mathbb{P}^n$.
