  In the minimal supersymmetric standard model (MSSM) the bino-wino
coannihilation provides a feasible way to accommodate the observed cosmological
dark matter (DM) relic density. However, such a scenario usually predicts a
very small DM-nucleon scattering cross section that is below the neutrino
floor, and can not be tested by DM direct detection experiments. In this work,
we investigate the discovery potential of this bino-wino co-annihilation region
by searching for the soft dilepton events from the process $pp \to \chi^0_2
(\to \ell^+\ell^- \chi^0_1) \chi^\pm_1+jets$ at the LHC. We find that the mass
of the wino-like $\chi^0_2$ can be probed up to about 310 (230) GeV at
$2\sigma$ ($5\sigma$) level for an integrated luminosity ${\cal L}=300$
fb$^{-1}$. In the future HL-LHC with 3000 fb$^{-1}$ luminosity, the
corresponding mass limits can be pushed up to 430 (330) GeV.
