  We analyze a 37 pb$^{-1}$ data sample collected with the SND detector at the
VEPP-2000 $e^+e^-$ collider in the center-of-mass energy range 1.05--2.00 GeV
and present an updated measurement of the $e^+e^- \to \omega \pi^0 \to
\pi^0\pi^0\gamma$ cross section. In particular, we correct the mistake in
radiative correction calculation made in our previous measurement based on a
part of the data. The measured cross section is fitted with the vector meson
dominance model with three $\rho$-like states and used to test the conserved
vector current hypothesis in the $\tau^-\to\omega\pi^-\nu_{\tau}$ decay.
