  Precision measurements of the Higgs boson properties at the LHC provide
relevant constraints on possible weak-scale extensions of the Standard Model
(SM). In the context of the Minimal Supersymmetric Standard Model (MSSM) these
constraints seem to suggest that all the additional, non-SM-like Higgs bosons
should be heavy, with masses larger than about 400 GeV. This article shows that
such results do not hold when the theory approaches the conditions for
"alignment independent of decoupling", where the lightest CP-even Higgs boson
has SM-like tree-level couplings to fermions and gauge bosons, independently of
the non-standard Higgs boson masses. The combination of current bounds from
direct Higgs boson searches at the LHC, along with the alignment conditions,
have a significant impact on the allowed MSSM parameter space yielding light
additional Higgs bosons. In particular, after ensuring the correct mass for the
lightest CP-even Higgs boson, we find that precision measurements and direct
searches are complementary, and may soon be able to probe the region of
non-SM-like Higgs boson with masses below the top quark pair mass threshold of
350 GeV and low to moderate values of $\tan\beta$.
