We have used the metal/amorphous silicon/metal tunnel junction as a model system to explore the role of localized states in electron transport through thin insulating layers. We measured the tunneling conductance as a function of temperature T, bias voltage V, and barrier thickness d. The data show marked deviations from the classical WKB tunneling theory in the limit of low T and V with d intermediate between the decay length in the barrier and the Mott variable range hopping length. The data are instead consistent with directed inelastic hopping along statistically rare but highly conductive ``chains'' of localized states. The most effective chains for a given set of conditions (T,V,d) contain a definite number of localized states, N>1, configured in a nearly optimal way in space and energy. The conductance of the lowest-order hopping channel (all chains with N=2) exhibits the characteristic voltage and temperature dependences ${\mathit{G}}_{2}^{\mathrm{hop}}$(V)\ensuremath{\propto}${\mathit{V}}^{4/3}$, and ${\mathit{G}}_{2}^{\mathrm{hop}}$(T)\ensuremath{\propto}${\mathit{T}}^{4/3}$, respectively, as predicted by theory. Higher-order channels (N>2) also conform to the theoretical predictions remarkably well. The physical nature of these highly conductive channels and their implications for conduction through thick tunnel barriers and thin dielectrics is discussed.