  We study electron-positron pair production by the combination of a strong,
constant electric field and a thermal background. We show that this process is
similar to dynamically assisted Schwinger pair production, where the strong
field is instead assisted by another coherent field, which is weaker but
faster. We treat the interaction with the photons from the thermal background
perturbatively, while the interaction with the electric field is
nonperturbative (i.e. a Furry picture expansion in $\alpha$). At
$\mathcal{O}(\alpha^2)$ we have ordinary perturbative Breit-Wheeler pair
production assisted nonperturbatively by the electric field. Already at this
order we recover the same exponential part of the probability as previous
studies, which did not expand in $\alpha$. This means that we do not have to
consider higher orders, so our approach allows us to calculate the
pre-exponential part of the probability, which has not been obtained before in
this regime. Although the prefactor is in general subdominant compared to the
exponential part, in this case it can be important because it scales as
$\alpha^2\ll1$ and is therefore much smaller than the prefactor at
$\mathcal{O}(\alpha^0)$ (pure Schwinger pair production). We show that, because
of the exponential enhancement, $\mathcal{O}(\alpha^2)$ still gives the
dominant contribution for temperatures above a certain threshold, but, because
of the small prefactor, the threshold is higher than what the exponential alone
would suggest.
