  One major obstacle in applications of Stein's method for compound Poisson
approximation is the availability of so-called magic factors (bounds on the
solution of the Stein equation) with favourable dependence on the parameters of
the approximating compound Poisson random variable. In general, the best such
bounds have an exponential dependence on these parameters, though in certain
situations better bounds are available. In this paper, we extend the region for
which well-behaved magic factors are available for compound Poisson
approximation in the Kolmogorov metric, allowing useful compound Poisson
approximation theorems to be established in some regimes where they were
previously unavailable. To illustrate the advantages offered by these new
bounds, we consider applications to runs, reliability systems, Poisson mixtures
and sums of independent random variables.
