  Background: Challenging questions of the $\alpha$-particle optical-model
potential (OMP) are still pointed out by recent high-precision measurements of
$\alpha$-induced reaction data below the Coulomb barrier. Moreover, the
reliability of a previous OMP for $\alpha$-particles on nuclei within the mass
number range 45$\leq$$A$$\leq$209 has been recently proved for emitted $\alpha$
particles as well, but only in the case of proton-induced reactions on Zn
isotopes [Phys. Rev. C {\bf 91}, 064611 (2015), Paper I].
  Purpose: Analysis of most recent $(\alpha,\gamma)$ reaction data for Ge and
Zr isotopes, which provides an additional validation of the above-mentioned
potential, is related to a further account of $\alpha$-particle emission in
neutron-induced reactions on Zr isotopes, at the same time with a suitable
description of all competitive processes.
  Methods: A consistent parameter set, established or validated by independent
analysis of recent various data, particularly $\gamma$-ray strength functions,
have been involved within model calculation of the $(\alpha,\gamma)$ as well as
$(n,\alpha)$ reaction cross sections. The latter are part of the whole analysis
of the neutron activation of Zr isotopes, in order to avoid any error
compensation or latent ambiguity.
  Results: The aforesaid potential provides a consistent description of recent
$\alpha$-induced reaction data with no empirical rescaling factors of the
$\gamma$ and/or nucleon widths. On the other hand, its use leads to
underestimated predictions of the pre-equilibrium emission and statistical
models for the $(n,\alpha)$ reaction cross sections.
  Conclusions: An optical potential with a volume imaginary component seems to
be needed to describe the low-energy $\alpha$-particle evaporation, while only
surface absorption occurs in $\alpha$-induced reactions at similar energies.
