  Methanol, one of the simplest complex organic molecules in the Interstellar
Medium (ISM), has been shown to be present and extended in cold environments
such as starless cores. We aim at studying methanol emission across several
starless cores and investigate the physical conditions at which methanol starts
to be efficiently formed, as well as how the physical structure of the cores
and their surrounding environment affect its distribution. Methanol and
C$^{18}$O emission lines at 3 mm have been observed with the IRAM 30m telescope
within the large program "Gas phase Elemental abundances in Molecular CloudS"
(GEMS) towards 66 positions across 12 starless cores in the Taurus Molecular
Cloud. A non-LTE radiative transfer code was used to compute the column
densities in all positions. We then used state-of-the-art chemical models to
reproduce our observations. We have computed N(CH$_3$OH)/N(C$^{18}$O) column
density ratios for all the observed offsets, and two different behaviours can
be recognised: the cores where the ratio peaks at the dust peak, and the cores
where the ratio peaks with a slight offset with respect to the dust peak ($\sim
$10000 AU). We suggest that the cause of this behaviour is the irradiation on
the cores due to protostars nearby which accelerate energetic particles along
their outflows. The chemical models, which do not take into account irradiation
variations, can reproduce fairly well the overall observed column density of
methanol, but cannot reproduce the two different radial profiles observed. We
confirm the substantial effect of the environment onto the distribution of
methanol in starless cores. We suggest that the clumpy medium generated by
protostellar outflows might cause a more efficient penetration of the
interstellar radiation field in the molecular cloud and have an impact on the
distribution of methanol in starless cores.
