  Previous, large samples of quasar absorption spectra have indicated some
evidence for relative variations in the fine-structure constant
($\Delta\alpha/\alpha$) across the sky. However, they were likely affected by
long-range distortions of the wavelength calibration, so it is important to
establish a statistical sample of more reliable results, from multiple
telescopes. Here we triple the sample of $\Delta\alpha/\alpha$ measurements
from the Subaru Telescope which have been `supercalibrated' to correct for
long-range distortions. A blinded analysis of the metallic ions in 6
intervening absorption systems in two Subaru quasar spectra provides no
evidence for $\alpha$ variation, with a weighted mean of
$\Delta\alpha/\alpha=3.0\pm2.8_{\rm stat}\pm2.0_{\rm sys}$ parts per million
(1$\sigma$ statistical and systematic uncertainties). The main remaining
systematic effects are uncertainties in the long-range distortion corrections,
absorption profile models, and errors from redispersing multiple quasar
exposures onto a common wavelength grid. The results also assume that
terrestrial isotopic abundances prevail in the absorbers; assuming only the
dominant terrestrial isotope is present significantly lowers
$\Delta\alpha/\alpha$, though it is still consistent with zero. Given the
location of the two quasars on the sky, our results do not support the evidence
for spatial $\alpha$ variation, especially when combined with the 21 other
recent measurements which were corrected for, or resistant to, long-range
distortions. Our spectra and absorption profile fits are publicly available.
