  It is increasingly becoming difficult, within a broad class of supersymmetric
models, to satisfactorily explain the discrepancy between the measured
$(g-2)_\mu$ and its standard model prediction, and at the same time satisfy all
the other constraints. In this paper we propose a new scheme of gauge mediation
by introducing new soft supersymmetry breaking mass parameters for the Higgs
sector in a minimal setup containing only a pair of $({\bf 5} + \bar{\bf 5})$
messenger fields of SU(5). This enables us to explain the $(g-2)_\mu$
discrepancy while avoiding all the existing constraints. We also provide
possible dynamical origin of the new soft mass parameters. The wino and
higgsino weighing below 500 GeV constitute the smoking gun signal at the (high
luminosity) LHC.
