Orbital eccentricity encodes key information about compact-binary formation channels and astrophysical environments, making it a critical target for gravitational-wave (GW) inference. We present parameter-estimation (PE) analyses of GWs from eccentric compact binaries with the SEOBNRv6EHM waveform model. Using long, highly eccentric numerical-relativity waveforms as synthetic signals, we compare parameter recovery across state-of-the-art eccentric models. We find that SEOBNRv5EHM and TEOBResumS-Dal'i can yield biased estimates of eccentricity, masses, and spins in the most challenging configurations, while SEOBNRv6EHM significantly reduces these biases. Applying SEOBNRv6EHM to 27 GW events from the O1–O4 LIGO–Virgo–KAGRA observing runs—including binary black hole, neutron-star–black-hole, and binary neutron-star mergers—we identify five events with mild support for eccentricity over the quasi-circular precessing-spin hypothesis, with Bayes factors ${log}_{10}{ℬ}_{\mathrm{\text{QCP}}}^{\mathrm{\text{EAS}}}>0.5$. Since SEOBNRv6EHM is applicable to generic planar binaries, we reanalyze five high-mass events allowing for unbound initial conditions. For three of them—including GW190521, previously claimed to originate from a dynamical capture—a direct-capture configuration is comparable to, or marginally favored over, both the eccentric aligned-spin and quasi-circular precessing-spin hypotheses ($\log_{10}\mathcal{B}^{\rm unbound}_{\rm QCP} \approx 0.2$–$0.6$ for GW190521). The recovered configurations are, however, astrophysically unrealistic and cannot be confidently discriminated from highly eccentric bound orbits, so these results do not, by themselves, support an unbound origin for these events. SEOBNRv6EHM is three to five times faster in PE analyses than SEOBNRv5EHM, while improving waveform accuracy, enabling efficient, large-scale GW inference with eccentric waveforms.