A systematic approach to improve the short time dynamics for classical mapping treatments of nonadiabatic dynamics is developed. This approach is based on the Taylor expansion of time-dependent observables around t=0. By sampling initial conditions in a manner that renders accurate static moments of the electronic population, it is shown that the short time electronic population dynamics described by classical mapping approaches for nonadiabatic dynamics can be greatly improved. The approach is illustrated on the example of the spin-boson model. For this problem, the analysis of the expansion coefficients reveals why classical mapping approaches to nonadiabatic dynamics often perform much worse for energetically biased reactions than they do for reactions with zero bias. The analysis presented here not only allows for the improvement of short time (and often long time) behavior, but also points to a systematic way of accessing how accurate a given classical mapping approach should be for a given problem.