Swift J1357.2−0933 is a transient low-mass X-ray binary hosting a stellar-mass black hole. The source exhibits optical dips and very broad emission lines during both outburst and quiescence, which are thought to be the result of a high orbital inclination. We present phase-resolved spectroscopy obtained with the 10.4 m Gran Telescopio Canarias (GTC). The spectra focus on the Hβ spectral region during X-ray quiescence. The emission line is exceptionally broad (full width at half maximum, FWHM > 4000 Å), in agreement with previous studies focused on Hα. A two-Gaussian fit to the prominent double-peaked profile reveals a periodic variability in the centroid position of the line. We also produced a diagnostic diagram aimed at constraining additional orbital parameters. Together, they allow us to independently confirm the orbital period of the system using a new dataset obtained five years after the previous outburst. However, our estimates for both the systemic velocity and the radial velocity semi-amplitude of the black hole reveal larger values than those found in previous studies. We argue that this could be explained by the precession of the disc and the presence of a hotspot. We found evidence of a narrow inner core in the double-peaked Hβ emission profile. We studied its evolution across the orbit, finding that it is likely to result from the occultation of inner material by the outer rim bulge, further supporting the high orbital inclination hypothesis.