We investigate ways to produce the bifurcation observed in the stellar stream of the Sagittarius dwarf galaxy (Sgr). Our method consists of running N-body simulations of Sgr falling into the Milky Way for the last 3 Gyr, with added test particles on disk orbits that span a wide range of initial positions, energies, and angular momenta. We find that particles that end up in the faint branch are predominantly high-angular-momentum particles that can all originate from a single plane within the progenitor, nearly perpendicular both to the orbital plane of the progenitor and to the Milky Way stellar disk. Their original configuration at the start of the simulation corresponds to spiral features already present 3 Gyr ago, which could be, e.g., the result of a disk-like component being tidally perturbed, or the tidal tails of a satellite being disrupted within Sgr. We then run a simulation including the self-gravity of this disky component. Despite the remaining ambiguity of its origin, this disk component of the Sgr dwarf with spiral overdensities provides a first step toward a working model to reproduce the observed faint branch of the bifurcated Sgr stream.