It is shown how the single-turning-point singular-perturbation analysis of Bender and Sharp may be extended to multiple-turning-point problems. The methodology presented emphasizes the importance of combining a function-moments analysis together with high-temperature lattice expansions and associated Pade analysis. This approach has been previously developed by Handy for nonlinear kink and soliton solutions and will be referred to as lattice multiscale singular-perturbation theory (LMSPT). The formalism is developed in the context of one-dimensional polynomial potential systems and highlights recursive moment relations heretofore unappreciated in general. As such, for these kinds of systems, the LMSPT approach offers an alternative to the conventional WKB method; and is extendible to nonlinear systems.