Magnetars are isolated young neutron stars characterized by the most intense magnetic fields known in the universe. The origin of their magnetic field is still a challenging question. In situ magnetic field amplification by dynamo action is a promising process to generate ultra-strong magnetic fields in fast-rotating progenitors. However, it is unclear whether the fraction of progenitors harboring fast core rotation is sufficient to explain the entire magnetar population. To address this point, we propose a new scenario for magnetar formation, in which a slow-rotating proto-neutron star is spun up by the supernova fallback. We argue that this can trigger the development of the Tayler-Spruit dynamo while other dynamo processes are disfavored. Using previous works done on this dynamo and simulations to characterize the fallback, we derive equations modelling the coupled evolution of the proto-neutron star rotation and magnetic field. Their time integration for different fallback masses is successfully compared with analytical estimates of the amplification timescales and saturation value of the magnetic field. We find that the magnetic field is amplified within $20$ to $40$s after the core bounce, and that the radial magnetic field saturates at intensities $10^{14}-10^{15}$G, therefore spanning the full range of magnetar's dipolar magnetic fields. We also compare predictions of two proposed saturation mechanisms showing that magnetar-like magnetic fields can be generated for neutron star spun up to rotation periods $\lesssim8$ms and $\lesssim28$ms, corresponding to fallback masses $\gtrsim4\times10^{-2}{\rm M}_{\odot}$ and $\gtrsim10^{-2}{\rm M}_{\odot}$. Thus, our results suggest that magnetars can be formed from slow-rotating progenitors for fallback masses compatible with recent supernova simulations and leading to plausible initial rotation periods of the proto-neutron star.