A previous study we reported in this journal suggested that left and right-handers may differ in their patterns of lateralization for different language tasks (Woodhead et al. 2019 R. Soc. Open Sci. 6, 181801. (doi:10.1098/rsos.181801)). However, it had too few left-handers (N = 7) to reach firm conclusions. For this update paper, further participants were added to the sample to create separate groups of left- (N = 31) and right-handers (N = 43). Two hypotheses were tested: (1) that lateralization would be weaker at the group level in left-than right-handers; and (2) that left-handers would show weaker covariance in lateralization between tasks, supporting a two-factor model. All participants performed the same protocol as in our previous paper: lateralization was measured using functional transcranial Doppler sonography during six different language tasks, on two separate testing sessions. The results supported hypothesis 1, with significant differences in laterality between groups for four out of six tasks. For hypothesis 2, structural equation modelling showed that there was stronger evidence for a two-factor model in left than right-handers; furthermore, examination of the factor loadings suggested that the pattern of laterality across tasks may also differ between handedness groups. These results expand on what is known about the differences in laterality between left- and right-handers.