A numerical semigroup $S$ is an additive subsemigroup of the non-negative integers with finite complement, and the squarefree divisor complex of an element $m \in S$ is a simplicial complex $Δ_m$ that arises in the study of multigraded Betti numbers. We compute squarefree divisor complexes for certain classes numerical semigroups, and exhibit a new family of simplicial complexes that are occur as the squarefree divisor complex of some numerical semigroup element.