  In this paper, we search for the regions of the phenomenological minimal
supersymmetric standard model (pMSSM) parameter space where one can expect to
have moderate Higgs mixing angle ($\alpha$) with relatively light (up to 600
GeV) additional Higgses after satisfying the current LHC data. We perform a
global fit analysis using most updated data (till December 2014) from the LHC
and Tevatron experiments. The constraints coming from the precision
measurements of the rare b-decays $B_s \to \mu^+ \mu^-$ and $b\to s \gamma$ are
also considered. We find that low $M_{A}$ $(\lesssim 350)$ and high $\tan\beta$
$(\gtrsim 25)$ regions are disfavored by the combined effect of the global
analysis and flavour data. However, regions with Higgs mixing angle $\alpha
\sim$ 0.1 - 0.8 are still allowed by the current data. We then study the
existing direct search bounds on the heavy scalar/pseudoscalar ($\rm H/A$) and
charged Higgs boson ($\rm H^\pm$) masses and branchings at the LHC. It has been
found that regions with low to moderate values of $\tan\beta$ with light
additional Higgses (mass $\le$ 600 GeV) are unconstrained by the data, while
the regions with $\tan\beta >$ 20 are excluded considering the direct search
bounds by the LHC-8 data. The possibility to probe the region with $\tan\beta$
$\le$ 20 at the high luminosity run of LHC are also discussed, giving special
attention to the $\rm H \to hh$, $\rm H/{A} \to t \bar t$ and $\rm H/{A} \to
\tau^{+}\tau^{-}$ decay modes.
