  In this note, we extend work of Farkas and Rim\'anyi on applying quadric rank
loci to finding divisors of small slope on the moduli space of curves by
instead considering all divisorial conditions on the hypersurfaces of a fixed
degree containing a projective curve. This gives rise to a large family of
virtual divisors on $\overline{\mathcal{M}_g}$. We determine explicitly which
of these divisors are candidate counterexamples to the Slope Conjecture. The
potential counterexamples exist on $\overline{\mathcal{M}_g}$, where the set of
possible values of $g\in \{1,\ldots,N\}$ has density $\Omega(\log(N)^{-0.087})$
for $N>>0$. Furthermore, no divisorial condition defined using hypersurfaces of
degree greater than 2 give counterexamples to the Slope Conjecture, and every
divisor in our family has slope at least $6+\frac{8}{g+1}$.
