  Let $X$ be a very general degree $d\geq 5$ hypersurface in $\mathbb{P}^3$. We
compute the ample cone of the Hilbert scheme $X^{[n]}$ of $n$ points on $X$ for
various small values of $n$ (the answer is already known for large $n$). We
obtain complete answers in some cases and find lower bounds in certain others.
We also observe that in the case of $X^{[2]}$ for quintic hypersurfaces $X$,
the existence (or absence) of hyperplane sections with points of high
multiplicity also plays a role in the answer to the question at hand, in
contrast with cases known earlier. Finally, in the case that a degree $d\geq3$
smooth hypersurface $X$ contains a line, we compute the nef cone of $X^{[n]}$
in a slice of the N\'eron-Severi space.
