  We discuss the onset of the dominance of the Glauber-Gribov-Landshoff (GGL)
component of pQCD hard two-gluon (2G) exchange contribution to hard elastic
$\pi\pi$ scattering at moderate energies . Such a hard $\pi\pi$ scattering
could via final state interaction on the $\gamma \gamma \to \pi^+ \pi^-$
reaction in which pQCD quark-exchange contribution is known to be short of
strength. While in the nonrelativistic approximation the GGL amplitude is known
to be free of suppression by the pion form factor, we show that in the
relativistic light-cone approach it acquires a residual, albeit a weak,
suppression. Furthermore,z the same mechanism it is free of the end-point
contributions. We evaluation the GGL amplitude with a model light-cone wave
function consistent with the pion charge form factor data. The soft
contribution to elastic $\pi\pi$ scattering is estimated based on the $NN$ and
$\pi N$ total cross section data and Regge factorization, which gives the
$\pi\pi$ total cross sections consistent with the ones deduced earlier from the
absorption model analysis of the $\pi N \to X N, X \Delta$ data. We evaluate
the large-$|t|$ tail of the soft amplitude within the Regge absorption models.
We find that while in the same sign $\pi^\pm\pi^\pm$ scattering the hard GGL
mechanism takes over at $|t|\gsim$ 3 GeV$^{-2}$, in the opposite-sign
$\pi^\pm\pi^\mp$ scattering the hard GGL mechanism $|t|\lsim$ 4 GeV$^{-2}$.
