Arguably the main challenge of galactic magnetism studies is to explain how the interstellar medium of galaxies reaches energetic equipartition despite the extremely weak cosmic primordial magnetic fields that are originally predicted to thread the inter-galactic medium. Previous numerical studies of isolated galaxies suggest that a fast dynamo amplification might suffice to brgalaxyidge the gap spanning many orders of magnitude in strength between the weak early Universe magnetic fields and the ones observed in high-redshift galaxies. To better understand their evolution in the cosmological context of hierarchical galaxy growth, we probe the amplification process undergone by the cosmic magnetic field within a spiral galaxy to unprecedented accuracy by means of a suite of constrained transport magnetohydrodynamical adaptive mesh refinement cosmological zoom simulations with different stellar feedback prescriptions. A galactic turbulent dynamo is found to be naturally excited in this cosmological environment, being responsible for most of the amplification of the magnetic energy. Indeed, we find that the magnetic energy spectra of simulated galaxies display telltale inverse cascades. Overall, the amplification process can be divided in three main phases, which are related to different physical mechanisms driving galaxy evolution: an initial collapse phase, an accretion-driven phase, and a feedback-driven phase. Whilst different feedback models affect the magnetic field amplification differently, all tested models prove to be sub-dominant at early epochs, before the feedback-driven phase is reached. Thus, the three-phase evolution paradigm is found to be quite robust vis-à-vis feedback prescriptions.