  We revisit the electroweak phase transition in the scalar singlet extension
of the standard model with a $\mathbb{Z}_2$ symmetry. In significant parts of
the parameter space the phase transition occurs in two steps - including
canonical benchmarks used in experimental projections for gravitational waves.
Domain walls produced in the first step of the transition seed the final step
to the electroweak vacuum, an effect which is typically neglected but leads to
an exponentially enhanced tunnelling rate. We improve previous results obtained
for the seeded transition, which made use of the thin-wall or high temperature
approximations, by using the mountain pass algorithm that was recently proposed
as a useful tool for seeded processes. We then determine the predictions of the
seeded transition for the latent heat, bubble size and characteristic time
scale of the transition. Differences compared to homogeneous transitions are
most pronounced when there are relatively few domain walls per hubble patch,
potentially leading to an enhanced gravitational wave signal. We also provide a
derivation of the percolation criteria for a generic seeded transition, which
applies to the domain wall seeds we consider as well as to strings and
monopoles.
