  We present a new set of index-based measurements of [$\alpha$/Fe] for a
sample of 2093 galaxies in the SAMI Galaxy Survey. Following earlier work, we
fit a global relation between [$\alpha$/Fe] and the galaxy velocity dispersion
$\sigma$ for red sequence galaxies,
[$\alpha$/Fe]=(0.378$\pm$0.009)log($\sigma$/100)+(0.155$\pm$0.003). We observe
a correlation between the residuals and the local environmental surface
density, whereas no such relation exists for blue cloud galaxies. In the full
sample, we find that elliptical galaxies in high-density environments are
$\alpha$-enhanced by up to 0.057$\pm$0.014 dex at velocity dispersions
$\sigma$<100 km/s, compared with those in low-density environments. This
$\alpha$-enhancement is morphology-dependent, with the offset decreasing along
the Hubble sequence towards spirals, which have an offset of 0.019$\pm$0.014
dex. At low velocity dispersion and controlling for morphology, we estimate
that star formation in high-density environments is truncated $\sim1$ Gyr
earlier than in low-density environments. For elliptical galaxies only, we find
support for a parabolic relationship between [$\alpha$/Fe] and $\sigma$, with
an environmental $\alpha$-enhancement of at least 0.03 dex. This suggests
strong contributions from both environment and mass-based quenching mechanisms.
However, there is no evidence for this behaviour in later morphological types.
