  Precise measurement of the Electric Dipole Moment (EDM) and anomalous
magnetic moment ($g$-2) of particles are important tests of Beyond Standard
Model (BSM) physics. It is generally believed that the tau lepton couples more
strongly to BSM due to its large mass, and can be searched for at collider
experiments. A new method to approximately reconstruct the neutrinos from the
hadronic decays of $\tau^-\tau^+$ pairs produced at $e^-e^+$ tau factories is
proposed. With all final state particle momenta available, observables based on
matrix elements and sensitive to BSM are calculated. It is estimated that with
50 ab$^{-1}$ of data to be delivered by the $Belle$-II experiment, a tau EDM
search with a 1-$\sigma$ level precision of $|d_\tau^{NP}|<2.04\times 10^{-19}$
e$\cdot$cm, and $g$-2 search with $|a_\tau^{NP}|<1.75\times 10^{-5}$ ($1.5\%$
of the SM prediction), can be expected when systematics are not considered. The
new precision can effectively constrain BSM models with heavy mirror neutrinos.
It can also constrain models containing a light scalar with mass at $O$(1 GeV),
which can explain the current muon $g$-2 anomaly as well. The method in this
work offers a new opportunity to search for BSM at current and future tau
factories with high precision.
