  (Abridged) Based on the rapidly increasing all-sky data of Faraday rotation
measures and polarised synchrotron radiation, the Milky Way's magnetic field is
now modelled with an unprecedented level of detail and complexity. We aim to
complement this heuristic approach with a physically motivated, quantitative
Galactic dynamo model -- a model that moreover allows for the evolution of the
system as a whole, instead of just solving the induction equation for a fixed
static disc. Building on the framework of mean-field magnetohydrodynamics and
extending it to the realm of a hybrid evolution, we perform three-dimensional
global simulations of the Galactic disc. Closure coefficients embodying the
mean-field dynamo are calibrated against resolved box simulations of
supernova-driven interstellar turbulence. The emerging dynamo solutions
comprise a mixture of the dominant axisymmetric S0 mode, with even parity, and
a subdominant A0 mode, with odd parity. Notably, such a superposition of modes
creates a strong localised vertical field on one side of the Galactic disc. We
moreover find significant radial pitch angles, which decay with radius --
explained by flaring of the disc. In accordance with previous work, magnetic
instabilities appear to be restricted to the less-stirred outer Galactic disc.
Their main effect is to create strong fields at large radii such that the
radial scale length of the magnetic field increases from 4 kpc (for the case of
a mean-field dynamo alone) to about 10 kpc in the hybrid models. There remain
aspects (e.g., spiral arms, X-shaped halo fields, fluctuating fields) that are
not captured by the current model and that will require further development
towards a fully dynamical evolution. Nevertheless, the work presented
demonstrates that a hybrid modelling of the Galactic dynamo is feasible and can
serve as a foundation for future efforts.
