We propose a hierarchical approach for the stochastic stability analysis of evolutionary dynamics. Each layer in the hierarchy represents a compromise between computational effort and the resolution of information about the long-run behavior of evolutionary dynamics. Previously, we proposed a graphical reformulation of Evolutionarily Sable Strategy (ESS) analysis through which we identified a set of strategies that cannot be ESS. Moreover, we also computed a set of strategies that was guaranteed to contain the set of stochastically stable strategies. The previous analysis was developed by considering transitions resulting from single mutations only. We extend the graphical approach to higher order analysis by incorporating mutations of higher order and show that we can refine our solution estimate by identifying smaller subsets of strategies that contain the set of stochastically stable strategies. However, this refinement comes at a cost of increase in computational budget.