Asymmetries in hand movements have routinely been attributed to properties of the two cerebral hemispheres. In right-handed participants, the non-dominant left hand tends to have shorter reaction times, with the dominant right hand achieving shorter movement durations as well as higher peak velocities. The root cause of the surprising left hand RT effect has been debated, largely in the context of right hemisphere specialisation in attention, visuospatial abilities, or “premotor” processes. Mieschke et al. (Brain Cognit 45:1, 2001) and Barthélémy and Boulinguez ( Behav Brain Res 133:1, 2002) both tried to dissociate “premotor” processes explaining the left hand RT advantage, using reaching paradigms where at least one condition required target detection, but no visually guided aiming movement. Unfortunately, the studies obtained conflicting results and conclusions. In the present study, we attempted to re-examine this kind of paradigm with methodological improvements, such as using a task with higher visuospatial demands. Our results demonstrate that whilst RTs are longer as movement complexity increases across three conditions, the left hand RT advantage is present across all conditions—and no significant interaction between hand and condition was found. No significant hand differences were found in peak velocity or duration. These results suggest that the left hand RT advantage cannot be due to movement planning advantages of the right hemisphere, and instead should be attributed to sustained attention/vigilance lateralisation to the right cerebral hemisphere.