  Due to their almost featureless optical/UV spectra, it is challenging to
measure the redshifts of BL Lacs. As a result, about 50% of gamma-ray BL Lacs
lack a firm measurement of this property, which is fundamental for population
studies, indirect estimates of the EBL, and fundamental physics probes. This
paper is the third in a series of papers aimed at determining the redshift of a
sample of blazars selected as prime targets for future observations with the
next generation, ground-based VHE gamma-ray astronomy observatory, Cherenkov
Telescope Array Observatory (CTAO). The accurate determination of the redshift
of these objects is an important aid in source selection and planning of future
CTAO observations. The selected targets were expected to be detectable with
CTAO in observations of 30 hours or less. We performed deep spectroscopic
observations of 41 of these blazars using the Keck II, Lick, SALT, GTC, and
ESO/VLT telescopes. We carefully searched for spectral lines in the spectra and
whenever features of the host galaxy were detected, we attempted to model the
properties of the host galaxy. The magnitudes of the targets at the time of the
observations were also compared to their long-term light curves. Spectra from
24 objects display spectral features or a high S/N. From these, 12
spectroscopic redshifts were determined, ranging from 0.2223 to 0.7018.
Furthermore, 1 tentative redshift (0.6622) and 2 redshift lower limits at z >
0.6185 and z > 0.6347 were obtained. The other 9 BL Lacs showed featureless
spectra, despite the high S/N (> 100) observations. Our comparisons with
long-term optical light curves tentatively suggest that redshift measurements
are more straightforward during an optical low state of the AGN. Overall, we
have determined 37 redshifts and 6 spectroscopic lower limits as part of our
programme thus far.
