Inspired by work of Altmann and Christophersen, we study the graded pieces of the cotangent cohomology $T^i_{S_{\mathcal{K}}}$, $i\geq 3$ of the Stanley-Reisner ring $S_{\mathcal{K}}$ associated to a simplicial complex $\mathcal{K}$. We prove a localization formula allowing one to reduce to the case of negative weights. Our results give a complete description of $T^3$ and $T^4$ in terms of the topology of $\mathcal{K}$ whenever $\mathcal{K}$ is a flag complex. As an application, we give a sufficient criterion for the vanishing of $T^3$ for simplicial spheres, classify two-spheres that have vanishing $T^3$, and show that the boundary complex of the dual associahedron has vanishing $T^3$. Our results make use of the arborescent resolutions considered by Hancharuk, Laurent-Gengoux, and Strobl. We give an alternative and self-contained treatment of these resolutions that may be of independent interest.