  Disc galaxies like the Milky Way are expected to be surrounded by massive
coronae of hot plasma that may contain a significant fraction of the so-called
missing baryons. We investigate whether the local (|vLSR|<400 km/s) warm-hot
absorption features observed towards extra-Galactic sources or halo stars are
consistent with being produced by the cooling of the Milky Way's corona. In our
scheme, cooling occurs at the interface between the disc and the corona and it
is triggered by positive supernova feedback. We combine hydrodynamical
simulations with a dynamical 3D model of the galactic fountain to predict the
all-sky distribution of this cooling material, and we compare it with the
observed distribution of detections for different `warm' (SiIII, SiIV, CII,
CIV) and `hot' (OVI) ionised species. The model reproduces the
position-velocity distribution and the column densities of the vast majority of
warm absorbers and about half of OVI absorbers. We conclude that the warm-hot
gas responsible for most of the detections lies within a few kiloparsecs from
the Galactic plane, where high-metallicity material from the disc mixes
efficiently with the hot corona. This process provides an accretion of a few
Mo/yr of fresh gas that can easily feed the star formation in the disc of the
Galaxy. The remaining OVI detections are likely to be a different population of
absorbers, located in the outskirts of the Galactic corona and/or in the
circumgalactic medium of nearby galaxies.
