  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.
