  A planet's orbital eccentricity is fundamental to understanding the present
dynamical state of a system and is a relic of its formation history. There is
high scientific value in measuring eccentricities of Kepler and TESS planets
given the sheer size of these samples and the diversity of their planetary
systems. However, Kepler and TESS lightcurves typically only permit robust
determinations of planet-to-star radius ratio $r$, orbital period $P$, and
transit mid-point $t_0$. Three other orbital properties, including impact
parameter $b$, eccentricity $e$, and argument of periastron $\omega$, are more
challenging to measure because they are all encoded in the lightcurve through
subtle effects on a single observable -- the transit duration $T_{14}$. In
Gilbert, MacDougall, & Petigura (2022), we showed that a five-parameter transit
description $\{P, t_0, r, b, T_{14}\}$ naturally yields unbiased measurements
of $r$ and $b$. Here, we build upon our previous work and introduce an accurate
and efficient prescription to measure $e$ and $\omega$. We validate this
approach through a suite of injection-and-recovery experiments. Our method
agrees with previous approaches that use a seven-parameter transit description
$\{P, t_0, r, b, \rho_\star, e, \omega\}$ which explicitly fits the
eccentricity vector and mean stellar density. The five-parameter method is
simpler than the seven-parameter method and is "future-proof" in that posterior
samples can be quickly reweighted (via importance sampling) to accommodate
updated priors and updated stellar properties. This method thus circumvents the
need for an expensive reanalysis of the raw photometry, offering a streamlined
path toward large-scale population analyses of eccentricity from transit
surveys.
