  The exotic decay modes of the already discovered 125-GeV scalar into a pair
of light pseudoscalars is a good probe of those new physics scenarios where
such pseudoscalars exist. Searches in the mass region where the pseudoscalar
($A$) is lighter than 62.5 GeV have yielded null findings so far. No search has
yet examined $m_A > 62.5$ GeV where the cross section is suppressed by the
off-shell pseudoscalar. We point out a possible enhancement of the sensitivity
of probing $hAA$ coupling in the context of a Type-X two Higgs doublet model.
We focus on $h\to A A^{(*)} \to 4\tau$, and select events with two same-sign
$\tau$-jets along with a pair of same-sign leptons. This enables much more
effective background elimination than in the erstwhile proposed channels.
Taking two values of the $\tau\tau A$ coupling into account, we obtain limits
on the $hAA$ coupling that can be probed at 2$\sigma$ and 3$\sigma$
significance, for $m_A$ ranging up to 85 GeV. For $m_A < 62.5$ GeV, too, we
find the probe through our suggested channel exhibits considerable improvement
upon the usual $2\mu 2\tau$-based searches conducted at the LHC. Within this
region, achieving the reach of $\lambda_{hAA}$ coupling at 3000 fb$^{-1}$
luminosity using our strategy would require approximately $\sim$ 3.6$\times
10^{5}$ fb$^{-1}$ luminosity using conventional $2\mu 2\tau$-based searches.
