Recent studies suggest spectroscopic differences explain a fraction of the variation in Type Ia supernova (SN Ia) luminosities after light-curve/color standardization. In this work, (i) we empirically characterize the variations of standardized SN Ia luminosities, and (ii) we use a spectroscopically inferred parameter, SIP, to improve the precision of SNe Ia along the distance ladder and the determination of the Hubble constant (H₀). First, we show that thePantheon+ covariance model modestly overestimates the uncertainty of standardized magnitudes by ∼ 7%, in the parameter space used by the SH0ES Team to measure H0; accounting for this alone yields H₀ = 73.01 ± 0.92 km s^-1 Mpc^-1. Furthermore, accounting for spectroscopic similarity between SNe Ia on the distance ladder reduces their relative scatter to ∼ 0.12 mag per object (compared to ∼ 0.14 mag previously). Combining these two findings in the model of SN covariance, we find an overall 14% reduction (to ± 0.85 km s^-1 Mpc^-1) of the uncertainty in the Hubble constant and a modest increase in its value. Including a budget for systematic uncertainties itemized by Riess et al. (2022a), we report an updated local Hubble constant with ∼ 1.2% uncertainty, H₀ = 73.29 ± 0.90 km s^-1 Mpc^-1. We conclude that spectroscopic differences among photometrically standardized SNe Ia do not explain the "Hubble tension". Rather, accounting for such differences increases its significance, as the discrepancy against ΛCDM calibrated by the Planck 2018 measurement rises to 5.7σ.