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→χ20(→ χ10)χ1±+jets at the LHC. We find that the mass of the winolike χ20 can be probed up to about 310 (230) GeV at 2σ (5σ) level for an integrated luminosity 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.