The search for primordial gravitational waves through the $B$-mode polarization pattern in the CMB is one of the major goals of current and future CMB experiments. Besides foregrounds, a potential hurdle in this search is the anisotropic secondary $B$-mode polarization generated by the scattering of CMB photons off free electrons produced during patchy cosmological reionization. Robust predictions of these secondary anisotropies are challenging because of uncertainties in the reionization history. In this paper, we revise estimates of the reionization-induced $B$-mode signal by incorporating recent advances in the understanding of reionization through observations of the Lyman-$alpha$ forest. To derive these $B$-mode estimates, we use high-dynamic-range radiative transfer simulations of reionization that are calibrated to the Ly$alpha$ data. These simulations are also consistent with a variety of other high-redshift observations. We find that around multipoles $ellapprox 100$, reionization induces $B$-mode power with $ell(ell+1)C_ell^{BB}/2piapprox 4times 10^{-6},mu$K$^2$. This secondary signal is thus at the level of the primordial signal with the tensor-to-scalar ratio $r<10^{-4}$, and can increase by a factor of $sim 50$ if reionization is sourced by highly clustered sources residing in haloes with mass of $sim 10^{11}$ M$_odot$. Our findings suggest that the contribution of patchy reionization to the search for primordial gravitational waves is unlikely to be a concern for currently planned CMB experiments.
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