A correlation between astrophysical high-energy neutrinos and blazars has been suggested by various authors. In particular, a likely association between IceCube events and intermediate- and high-energy peaked BL Lac objects has led to a sample of 47 objects having a high probability of being neutrino sources. In the first paper of this series we reported optical spectroscopy of 17 objects, which together with data taken from the literature covered 80 per cent of the sample. Here, we present spectroscopy obtained at large aperture telescopes of a further 17 objects (plus four additional targets), which completes the sample coverage. For 12 objects we are able to determine the redshift (0.07 < z < 1.6), while for the others we set a lower limit on it, based on either the robust detection of intervening absorption systems or on an estimation derived from the absence of spectral signatures of the host galaxy. With these new data we expand and reinforce the main results of our previous papers, namely the fact that in terms of their broad-band properties our sources appear to be indistinguishable from the rest of the blazar population and the relatively large (>34 per cent and possibly as high as 80 per cent) fraction of masquerading BL Lac objects, for which the low equivalent width of the emission lines is due to the brightness of the boosted continuum, rather than being an intrinsic property, in our sample.