  We show how the conic hull method, recently developed for the analytic and
non-iterative evaluation of multifold Mellin-Barnes (MB) integrals, can be
extended to the case where these integrals have straight contours of
integration parallel to the imaginary axes in the complex planes of the
integration variables. MB integrals of this class appear, for instance, when
one computes the $\epsilon$-expansion of dimensionally regularized Feynman
integrals, as a result of the application of one of the two main strategies
(called A and B in the literature) used to resolve the singularities in
$\epsilon$ of MB representations. We upgrade the Mathematica package
MBConicHulls.wl which can now be used to obtain multivariable series
representations of multifold MB integrals with arbitrary straight contours,
providing an efficient tool for the automatic computation of such integrals.
This new feature of the package is presented, along with an example of
application by calculating the $\epsilon$-expansion of the dimensionally
regularized massless one-loop pentagon integral in general kinematics and
$D=4-2\epsilon$.
