Context. Blazars, comprising BL Lacertae objects (BL Lacs) and flat-spectrum radio quasars, are the most luminous extragalactic sources that dominate the γ-ray sky. They account for approximately 56% of the sources listed in the recent Fermi-LAT (Large Area Telescope) catalogue (4FGL-DR4). The optical and UV spectra of BL Lacs are nearly featureless, making it difficult to precisely determine their redshifts. Consequently, nearly half of the γ-ray BL Lacs lack reliable redshift measurements. This poses a significant challenge, since redshift is crucial for studying the cosmic evolution of the blazar population and for understanding their intrinsic emission mechanisms. Additionally, it is vital for γ-ray propagation studies, such as indirect evidence of extragalactic background light (EBL), placing constraints on the intergalactic magnetic field (IGMF), and searches for Lorentz invariance violation (LIV) and axion-like particles (ALPs). Aims. This paper is the fourth in a series dedicated to determining the redshift of a sample of blazars identified as key targets for future observations with the Cherenkov Telescope Array Observatory (CTAO). The precise determination of the redshifts of these objects plays a crucial role in planning future CTAO observations. Methods. We carried out Monte Carlo simulations to identify potential γ-ray blazars with hard spectra detected by the Fermi-LAT telescope that currently lack redshift measurements. These simulations selected the blazars that are anticipated to be detectable by the CTAO within 30 hours or less of exposure assuming an average flux state. In this fourth paper, we report the results of detailed spectroscopic observations of 29 blazars using the ESO/VLT, Keck II, and SALT telescopes. Our analysis involved a thorough search for spectral lines in the spectra of each blazar, and when features of the host galaxy were identified, we modelled its properties. Moreover, we compared the magnitudes of the targets during the observations to their long-term light curves. Results. In the sample studied, 9 of 29 sources were observed with a high signal-to-noise ratio (S/N > 100), while the remaining 20 were observed with a moderate or low S/N. We successfully determined firm redshifts for 12 blazars, ranging from 0.1636 to 1.1427, and identified two lower limit redshifts at z > 1.0196 and z > 1.4454. The remaining 15 BL Lac objects exhibited featureless spectra under the observed S/N.