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. (read more)