Abstract Ly α emission from galaxies can be used to trace neutral hydrogen in the epoch of reionization, however, there is a degeneracy between the attenuation of Ly α in the intergalactic medium (IGM) and the line profile emitted by the galaxy. Large shifts of Ly α redward of systemic due to scattering in the interstellar medium can boost Ly α transmission in the IGM during reionization. The relationship between the Ly α velocity offset from systemic and other galaxy properties is not well established at high redshift or low luminosities, due to the difficulty of observing emission lines which trace the systemic redshift. Rest-frame optical spectroscopy with JWST/NIRSpec has opened a new window into understanding Ly α at z > 3. We present a sample of 12 UV-faint galaxies (−20 ≲ M UV ≲ −16) at 3 ≲ z ≲ 6, with Ly α velocity offsets, Δ v Ly α , measured from the Very Large Telescope/MUSE and JWST/NIRSpec from the GLASS-JWST Early Release Program. We find a median Δ v Ly α of 205 km s −1 and standard deviation of 75 km s −1 , compared to 320 and 170 km s −1 , respectively, for M UV < −20 galaxies in the literature. Our new sample demonstrates the previously observed trend of decreasing Ly α velocity offset with decreasing UV luminosity and optical line velocity dispersion, which extends to M UV ≳ −20, consistent with a picture where the Ly α profile is shaped by gas close to the systemic redshift. Our results imply that during reionization Ly α from UV-faint galaxies will be preferentially attenuated, but that detecting Ly α with low Δ v Ly α can be an indicator of large ionized bubbles.