In the Minimal Supersymmetric Standard Model (MSSM) searches for the heaviest CP-even and CP-odd Higgs $H$, $A$ to tau-lepton pairs severely constrain the parameter region for large values of $\tan\beta$ and light Higgs bosons $H$, $A$. We demonstrate how the experimental constraint can be avoided by new decays to light third-generation sfermions, whose left-right couplings to $H$ can be maximised in regions of large trilinear couplings. We also show that a new search strategy for the discovery of staus at the LHC is possible in the case of stau-pair production via $H/A$ decay, which can be between one or two orders of magnitude larger than the usual electroweak cross-section production for staus.