  Using a novel optimization procedure it has been shown that the Helically
Symmetric eXperiment (HSX) stellarator can be optimized for reduced
trapped-electron-mode (TEM) instability [M.J.~Gerard et al.,
\textit{Nucl.~Fusion} \textbf{63} (2023) 056004]. Presently, with a set of 563
experimental candidate configurations, gyrokinetic simulations are performed to
investigate the efficacy of available energy $E_\mathrm{A}$, quasi-helical
symmetry, and flux-surface shaping parameters as metrics for TEM stabilization.
It is found that lower values of $E_\mathrm{A}$ correlate with reduced growth
rates, but only when separate flux-surface shaping regimes are considered.
Moreover, configurations with improved quasi-helical symmetry demonstrate a
similar reduction in growth rates and less scatter compared to $E_\mathrm{A}$.
Regarding flux-surface shaping, a set of helical shaping parameters is
introduced that show increased elongation is strongly correlated with reduced
TEM growth rates, however, only when the quasi-helical symmetry is preserved.
Using a newly derived velocity-space-averaged TEM resonance operator, these
trends are analyzed to provide insights into the physical mechanism of the
observed stabilization. For elongation, stabilization is attributed to
geometric effects that reduce the destabilizing particle drifts across the
magnetic field. Regarding quasi-helical symmetry, the TEM resonance in the
maximally resonant trapping well is shown to increase as the quasi-helical
symmetry is broken, and breaking quasi-helical symmetry increases the
prevalence of highly resonant trapping wells. While these results demonstrate
the limitations of using any single metric as a linear TEM proxy, it is shown
that quasi-helical symmetry and plasma elongation are highly effective metrics
for reducing TEM growth rates in helical equilibria.
