  One of the main implications of the LHC discovery of a Higgs boson with a
mass $M_h \approx 126$ GeV is that the scale of supersymmetry-breaking in the
Minimal Supersymmetric Standard Model (MSSM) might be rather high, $M_S \gg
M_Z$. In this paper, we consider the high $M_S$ regime and study the spectrum
of the extended Higgs sector of the MSSM, including the LHC constraints on the
mass and the rates of the observed light $h$ state. In particular, we show that
in a simplified model that approximates the important radiative corrections,
the unknown scale $M_S$ (and some other leading SUSY parameters) can be traded
against the measured value of $M_h$. One would be then essentially left with
only two free parameters to describe the Higgs sector, tan$\beta$ and the
pseudoscalar Higgs mass $M_A$, even at higher orders. The main phenomenological
consequence of these high $M_S$ values is to reopen the low tan$\beta$ region,
tan$\beta \lsim 3-5$, which was for a long time buried under the LEP constraint
on the lightest $h$ mass when a low SUSY scale was assumed. We show that, in
this case, the heavier MSSM neutral $H/A$ and charged $H^\pm$ states can be
searched for in a variety of interesting final states such as decays into gauge
and lighter Higgs bosons (in pairs on in mixed states) and decays into heavy
top quarks. Examples of sensitivity on the $[tan\beta, M_A]$ parameter space at
the LHC in these channels are given.
