We revisit Holstein's polaron model to derive an extension of the expression for the thermal dependence of the electrical resistivity in the non-adiabatic small-polaron regime. Our analysis relaxes Holstein's assumption that the vibrational-mode energies are much smaller than the thermal energy k (B) T and substitutes a fifth-order expansion in powers of for the linear approximation in the expression for the quasiparticle hopping probability in the original treatment. The resulting expression for the electrical resistivity has the form rho(T)=rho (0) T (3/2) exp(E (a) /k (B) T-C/T (3)+D/T (5)), where C and D are constants related to the molecule-electron interaction energy, or alternatively to the polaron binding energy, and the dispersion relation of the vibrational normal modes. We show that experimental data for the La (1-x) Ca (x) MnO (3) (x=0.30,0.34,0.40, and 0.45) manganite system, which are poorly fitted by the conventional non-adiabatic model, are remarkably well described by the more accurate expression. Our results suggest that, under conditions favoring high resistivity, the higher-order terms associated with the constants C and D in the above expression should taken into account in comparisons between theoretical and experimental results for the temperature-dependent transport properties of transition-metal oxides.