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 six intervening absorption systems in two Subaru quasar spectra provides no evidence for a variation, with a weighted mean of Delta alpha/alpha = 3.0 +/- 2.8(stat) +/- 2.0(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 on to 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 a 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.