  Due to shielding, direct detection experiments are in some cases insensitive
to dark matter candidates with very large scattering cross sections with
nucleons. In this paper, we revisit this class of models, and derive a simple
analytic criterion for conservative but robust direct detection limits. While
large spin-independent cross sections seem to be ruled out, we identify
potentially viable parameter space for dark matter with a spin-dependent cross
section with nucleons in the range of $10^{-27} {\rm cm}^2 < \sigma_{{\rm
DM}-p} < 10^{-24} \, {\rm cm}^{2}$. With these parameters, cosmic-ray
scattering with dark matter in the extended halo of the Milky Way could
generate a novel and distinctive gamma-ray signal at high galactic latitudes.
Such a signal could be observable by Fermi or future space-based gamma-ray
telescopes.
