In LEP times, hadronic tau decays were the most precise input for the (leading-order) hadronic vacuum polarization piece (HVP,LO) of the muon anomalous magnetic moment ($a_\mu$). With the advent of Φ- and B-factories, $e^+e^−$ hadronic cross-section surpassed them, giving the most accurate input for this piece. However, since both data-driven determinations are subject to theoretical uncertainties (isospin-breaking for taus and radiator function for dealing with initial-state radiation at the Monte Carlo level for $e^+e^−$) it is nevertheless needed to keep on improving the systematic theory uncertainties for using both data sets, as unprecedented precision is requested by the forthcoming measurement of $a_\mu$ at FNAL (that should be released before the end of this year). In this context, we have revisited the computations by Cirigliano et al. [Phys.Lett. B513 (2001) 361-370 and JHEP 0208 (2002) 002] to improve the SM theoretical uncertainty associated to the use of (the dominant) $\tau \to \pi \pi \nu_\tau$ decays for $a^{HVP,LO}_\mu$.