  We improve, in the energy region between $\bar{K}K$ threshold and $\sim~1.4$
GeV, the energy-dependent phase shift analysis of $\pi\pi$ scattering presented
in a previous paper. For the S0 wave we have included more data above
$\bar{K}K$ threshold and we have taken into account systematically the
elasticity data on the reaction $\pi\pi\to\bar{K}K$. We here made a coupled
channel fit. For the D0 wave we have considered information on low energy
parameters, and imposed a better fit to the $f_2$ resonance. For both waves the
expressions we now find are substantially more precise than the previous ones.
We also provide slightly improved D2 and P waves, including the estimated
inelasticity for the first, and a more flexible parametrization between 1 and
1.42 GeV for the second. The accuracy of our amplitudes is now such that it
requires a refinement of the Regge analysis, for $s^{1/2}\geq1.42$ GeV, which
we also carry out. We show that this more realistic input produces $\pi\pi$
scattering amplitudes that satisfy better forward dispersion relations,
particularly for $\pi^0\pi^0$ scattering.
