  We report a fairly detailed finite-size scaling analysis of the first-order
phase transition in the three-dimensional 3-state Potts model on cubic lattices
with emphasis on recently introduced quantities whose infinite-volume
extrapolations are governed `only' by exponentially small terms. In these
quantities no asymptotic power series in the inverse volume are involved which
complicate the finite-size scaling behaviour of standard observables related to
the specific-heat maxima or Binder-parameter minima. Introduced initially for
strong first-order phase transitions in q-state Potts models with ``large
enough'' q, the new techniques prove to be surprisingly accurate for a q value
as small as 3. On the basis of the high-precision Monte Carlo data of Alves `et
al.' [Phys. Rev. B43 (1991) 5846], this leads to a refined estimate of $\beta_t
= 0.550,565(10)$ for the infinite-volume transition point.
