  Magnetic fields are considered as a vital ingredient of contemporary star
formation, and may have been important during the formation of the first stars
in the presence of an efficient amplification mechanism. Initial seed fields
are provided via plasma fluctuations, and are subsequently amplified by the
small-scale dynamo, leading to a strong tangled magnetic field. Here we explore
how the magnetic field provided by the small-scale dynamo is further amplified
via the $\alpha-\Omega$ dynamo in a protostellar disk and assess its
implications. For this purpose, we consider two characteristic cases, a typical
Pop.~III star with $10$~M$_\odot$ and an accretion rate of
$10^{-3}$~M$_\odot$~yr$^{-1}$, and a supermassive star with $10^5$~M$_\odot$
and an accretion rate of $10^{-1}$~M$_\odot$~yr$^{-1}$. For the $10$~M$_\odot$
Pop.~III star, we find that coherent magnetic fields can be produced on scales
of at least $100$~AU, which are sufficient to drive a jet with a luminosity of
$100$~L$_\odot$ and a mass outflow rate of $10^{-3.7}$~M$_\odot$~yr$^{-1}$. For
the supermassive star, the dynamical timescales in its environment are even
shorter, implying smaller orbital timescales and an efficient magnetization out
to at least $1000$~AU. The jet luminosity corresponds to
$\sim10^{6.0}$~L$_\odot$, and a mass outflow rate of
$10^{-2.1}$~M$_\odot$~yr$^{-1}$. We expect that the feedback from the
supermassive star can have a relevant impact on its host galaxy.
