  We analyse the chances of detecting charged Higgs bosons of the Minimal
Supersymmetric Standard Model (MSSM) at the Large Hadron Collider (LHC) in the
$W^\pm h$ mode, followed by the dominant decay of the lightest Higgs scalar,
$h\to b\bar b$. If the actual value of $M_h$ is already known, this channel
offers possibly the optimal final state kinematics for charged Higgs discovery,
thanks to the narrow resonances appearing around the $W^\pm$ and $h$ masses.
Besides, within the MSSM, the $H^\pm\to W^\pm h$ decay rate is significant for
not too large $\tan\beta$ values, thus offering the possibility of accessing a
region of MSSM parameter space left uncovered by other search channels. We
consider both strong (QCD) and electroweak (EW) `irreducible' backgrounds in
the $3b$-tagged channel to the $gg\to t\bar bH^-$ production process that had
not been taken into account in previous analyses. After a series of kinematic
cuts, the largest of these processes is $t\bar bW^\pm h$ production in the
continuum. However, for optimum $\tan\beta$, i.e., between 2 and 3, the charged
Higgs boson signal overcomes this background and a narrow discovery region
survives around $M_{H^\pm}\approx200$ GeV
