  The neutralino dark matter (DM) predicted by the Minimal Supersymmetric
Standard Model (MSSM) has been probed in several search modes at the Large
Hadron Collider (LHC), one of the leading ones among which is the trilepton
plus missing transverse momentum channel. The experimental analysis of this
mode has, however, been designed to probe mainly a bino-like DM, originating in
the decays of a pair of next-to-lightest neutralino and lightest chargino, both
of which are assumed to be wino-like. In this study, we analyse how this
trilepton channel can be tuned for probing also the wino-like DM. We note that,
while the mentioned standard production mode generally leads to a relatively
poor sensitivity for the wino-like DM, there are regions in the MSSM parameter
space where the net yield in the trilepton final state can be substantially
enhanced at the LHC with $\sqrt{s}=14$ TeV. This is achieved by taking into
account also an alternative channel, pair-production of the wino-like DM itself
in association with the heavier chargino, and optimisation of the kinematical
cuts currently employed by the LHC collaborations. In particular, we find that
the cut on the transverse mass of the third lepton highly suppresses both the
signal channels and should therefore be discarded in this DM scenario. We
perform a detailed detector-level study of some selected parameter space points
that are consistent with the most important experimental constraints, including
the recent ones from the direct and indirect DM detection facilities. Our
analysis demonstrates the high complementarity of the two channels, with their
combined significance reaching above 4$\sigma$ for a wino-like DM mass around
100 GeV, with an integrated luminosity as low as 100 fb$^{-1}$.
