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.