  We present a general procedure for incorporating higher-order information
into the scale-setting prescription of Brodsky, Lepage and Mackenzie. In
particular, we show how to apply this prescription when the leading coefficient
or coefficients in a series in the strong coupling alpha_s are anomalously
small and the original prescription can give an unphysical scale. We give a
general method for computing an optimum scale numerically, within dimensional
regularization, and in cases when the coefficients of a series are known. We
apply it to the heavy quark mass and energy renormalization in lattice NRQCD,
and to a variety of known series. Among the latter, we find significant
corrections to the scales for the ratio of e+e- to hadrons over muons, the
ratio of the quark pole to MSbar mass, the semi-leptonic B-meson decay width,
and the top decay width. Scales for the latter two decay widths, expressed in
terms of MSbar masses, increase by factors of five and thirteen, respectively,
substantially reducing the size of radiative corrections.
