Supermassive black holes at the centre of galactic nuclei mostly grow in mass through gas accretion over cosmic time. This process also modifies the angular momentum (or spin) of black holes, both in magnitude and in orientation. Despite being often neglected in galaxy formation simulations, spin plays a crucial role in modulating accretion power, driving jet feedback, and determining recoil velocity of coalescing black hole binaries. We present a new accretion model for the moving-mesh code {\sc arepo} that incorporates (i) mass accretion through a thin $\alpha$-disc, and (ii) spin evolution through the Bardeen-Petterson effect. We use a diverse suite of idealised simulations to explore the physical connection between spin evolution and larger scale environment. We find that black holes with mass $\lesssim 107$ M$\mathrm{<mrow\; data-mjx-texclass="ORD">\odot </mrow>}$ experience quick alignment with the accretion disc. This favours prolonged phases of spin-up, and the spin direction evolves according to the gas inflow on timescales as short as $\lesssim 100$ Myr, which might explain the observed jet direction distribution in Seyfert galaxies. Heavier black holes ($\gtrsim 108$ M$\mathrm{<mrow\; data-mjx-texclass="ORD">\odot </mrow>}$) are instead more sensitive to the local gas kinematic. Here we find a wider distribution in spin magnitudes: spin-ups are favoured if gas inflow maintains a preferential direction, and spin-downs occur for nearly isotropic infall, while the spin direction does not change much over short timescales $\sim 100$ Myr. We therefore conclude that supermassive black holes with masses $\gtrsim 5\times 108$ M$\mathrm{<mrow\; data-mjx-texclass="ORD">\odot </mrow>}$ may be the ideal testbed to determine the main mode of black hole fuelling over cosmic time.