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 εij, neutrino evolution exhibits a significant time dependence. For |εττ|≳ 10−3 the neutrino survival probability may become sensitive to the θ23 octant and the sign of εττ. In particular, if εττ≳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 νe, instead of the usual spectral swap.