  In the context of the Minimal Supersymmetric extension of the Standard Model
(MSSM), we reanalyze the search for the heavier CP-even $H$ and CP-odd $A$
neutral Higgs bosons at the LHC in their production in the gluon-fusion
mechanism and their decays into gauge and lighter $h$ bosons and into top quark
pairs. We show that only when considering these processes, that one can fully
cover the entire parameter space of the Higgs sector of the model. Indeed, they
are sensitive to the low $\tan\beta$ and high Higgs mass ranges, complementing
the traditional searches for high mass resonances decaying into $\tau$-lepton
pairs which are instead sensitive to the large and moderate $\tan\beta$
regions. The complementarity of the various channels in the probing of the
complete $[\tan\beta, M_A]$ MSSM parameter space at the previous and upcoming
phases of the LHC is illustrated in a recently proposed simple and model
independent approach for the Higgs sector, the $h$MSSM, that we also refine in
this paper.
