  We consider the effect of non-standard neutrino interactions (NSI, for short)
on the propagation of neutrinos through the supernova (SN) envelope within a
three-neutrino framework and taking into account the presence of a neutrino
background. We find that for given NSI parameters, with strength generically
denoted by $\varepsilon_{ij}$, neutrino evolution exhibits a significant time
dependence. For $|\varepsilon_{\tau\tau}|\gtrsim$ $10^{-3}$ the neutrino
survival probability may become sensitive to the $\theta_{23}$ octant and the
sign of $\varepsilon_{\tau\tau}$. In particular, if
$\varepsilon_{\tau\tau}\gtrsim 10^{-2}$ an internal $I$-resonance may arise
independently of the matter density. For typical values found in SN simulations
this takes place in the same dense-neutrino region above the neutrinosphere
where collective effects occur, in particular during the synchronization
regime. This resonance may lead to an exchange of the neutrino fluxes entering
the bipolar regime. The main consequences are (i) bipolar conversion taking
place for normal neutrino mass hierarchy and (ii) a transformation of the flux
of low-energy $\nu_e$, instead of the usual spectral swap.
