  We continue our study of contour deformation as a practical tool for dealing
with the sign problem using the $d$-dimensional Bose gas with non-zero chemical
potential as a toy model. We derive explicit expressions for contours up to the
second order with respect to a natural small parameter and generalise these
contours to an ansatz for which the evaluation of the Jacobian is fast
($O(1)$). We examine the behaviour of the various proposed contours as a
function of space-time dimensionality, the chemical potential, and lattice size
and geometry and use the mean phase factor as a measure of the severity of the
sign problem. In turns out that this method leads to a substantial reduction of
the sign problem and that it becomes more efficient as space-time
dimensionality is increased. Correlations among contributions to
$\operatorname{Im}\langle S \rangle$ play a key role in determining the mean
phase factor and we examine these correlations in detail.
