We consider the set $\Sigma_0$ of smooth connected closed hypersufaces of contact type in $T^*N$ whose intersection with each fiber is either empty or bounds a compact region that contains the zero point. We show that if $N$ has dimension bigger than one and $H_1(N) = 0$ then there exists a subset $\widehat\Sigma_0 \subset \Sigma_0$ of symplectically non-convex hypersurfaces which is $C^1$-dense and open in the Hausdorff topology. The proof involves the Maslov index of periodic orbits and symplectic homology.