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In the Minimal Supersymmetric Standard Model (MSSM) searches for the heaviestCP-even and CP-odd Higgs $H$, $A$ to tau-lepton pairs severely constrain theparameter region for large values of $\tan\beta$ and light Higgs bosons $H$,$A$. We demonstrate how the experimental constraint can be avoided by newdecays to light third-generation sfermions, whose left-right couplings to $H$can be maximised in regions of large trilinear couplings $A_{b}$, $A_{\tau}$for sbottoms and staus, or large supersymmetric (SUSY) Higgs mass $\mu$ forstops. Due to the $\tan\beta$-enhancement in the production cross-sections viagluon-fusion and in association with bottom-quark pairs for $H$ and $A$, wefind that down-type sfermions, in particular, sbottoms perform a better job inallowing more parameter space than up-type sfermions such as stops, whichrequire much larger values of $\mu$ to compensate for $\tan\beta$. Vacuumstability as well as flavour observables constraints and direct searches forSUSY particles are imposed. We also associate the lightest CP-even Higgs withthe observed 125 GeV SM-like Higgs and impose the experimental constraints fromthe LHC.

Enlarging Regions of the MSSM Parameter Space for Large $\tan\beta$ via SUSY Decays of the Heavy Higgs Bosons