This paper presents a novel solution to the inverse Frobenius-Perron problem of reconstructing an unknown nonlinear and ergodic map from causal sequences of probability density functions generated by the map. The original solution to this problem successfully reconstructs members of the canonical map class (i.e., a subset of the piecewise linear semi-Markov maps), provided all the map’s branches are monotonically increasing. The original solution constructs a matrix estimate of the map’s Frobenius-Perron operator, which governs the evolution of density functions under iteration of the map, from the density sequences. The one-dimensional map is reconstructed from this matrix. In contrast, the proposed solution constructs a higher-order matrix estimate of the Frobenius-Perron operator. A member of the newly proposed class of generalized hat maps, a superset of the canonical maps, is constructed from this matrix estimate. The proposed solution successfully distinguishes between increasing and decreasing map branches and enlarges the class of maps that can be successfully reconstructed to canonical maps with any subset of decreasing branches. When used to reconstruct any piecewise linear semi-Markov map, the proposed solution generates a map with consistent invariant density and power spectrum mode characteristics, regardless of the unknown map’s canonicity or branch monotonicity. Numerical examples that illustrate the proposed solution’s characteristics are presented.