  We reconsider the issue of the search for a nonzero electric dipole form
factor (EDM) $d_\tau(s)$ using optimal observables in $\tau^+\tau^-$ production
by $e^+ e^-$ collisions in the center-of-mass energy range from the $\tau$-pair
threshold to about $\sqrt{s} \sim 15$ GeV. We discuss the general formalism of
optimal observables and apply it to two $CP$-odd observables that are sensitive
to the real and imaginary part of $d_\tau(s)$, respectively. We compute the
expectation values and covariances of these optimal $CP$ observables for
  $\tau$-pair production at $\sqrt{s}=10.58$ GeV with subsequent decays of
$\tau^\pm$ into major leptonic or semihadronic modes. For the $\tau$ decays to
two pions and three charged pions we take the full kinematic information of the
hadronic system into account. Assuming that the Belle II experiment at the KEKB
accelerator will eventually
  analyze data corresponding to an integrated luminosity of 50 ab$^{-1}$ and
applying acceptance cuts on the final-state pions we find that 1~s.d.
sensitivities $\delta {\rm Re} d_\tau = 6.8 \times 10^{-20}$ e cm and $\delta
{\rm} Im d_\tau = 4.0 \times 10^{-20}$ e cm can be obtained with events where
both $\tau$'s decay semihadronically.
  In the ideal case where no cuts on the final-state particles are applied we
find with 50 ab$^{-1}$ at $\sqrt{s}=10.58$ GeV corresponding to $4.5 \times
10^{10}$ $\tau^+ \tau^-$ events the 1 s.d. sensitivities $\delta {\rm Re}
d_\tau = 5.8 \times 10^{-20}$ e cm and $\delta {\rm} Im d_\tau = 3.2 \times
10^{-20}$ e cm, again for events where both $\tau$ leptons decay
semihadronically. Furthermore, we analyze the potential magnitude of the $\tau$
EDM form factor in the type-II two-Higgs doublet extension and in two scalar
leptoquark extensions of the Standard Model.
