  We reconsider the issue of the search for a non-zero 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 the $\Upsilon(4{\rm S})$ resonance 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 record and analyze $4.5
\times 10^{10}$ $\tau^+ \tau^-$ events at $\sqrt{s}=10.58$ GeV we find that 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 can be obtained with events
where both $\tau$'s 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, taking into account phenomenological constraints.
