Situation awareness (SA) describes an individual’s understanding of their surroundings and actions in the near future based on the individual’s comprehension and understanding of the surrounding inputs. SA measurements can be applied to improve system performance or human effectiveness in many fields of study, including driving. However, in some scenarios drivers might need to drive in unfamiliar traffic regulations (UFTRs), where the traffic rules and vehicle configurations are a bit different from what the drivers are used to under familiar traffic regulations. Such driving conditions require drivers to adapt their attention, knowledge, and reactions to safely reach the destination. This ability is influenced by the degree of handedness. In such tasks, mixed-/left-handed people show better performance than strong right-handed people. This paper aims to explore the influence of the degree of handedness on SA when driving under UFTRs. We analyzed the SA of two groups of drivers: strong right-handed drivers and mixed-/left-handed drivers. Both groups were not familiar with driving in keep-left traffic regulations. Using a driving simulator, all participants drove in a simulated keep-left traffic system. The participants’ SA was measured using a subjective assessment, named the Participant Situation Awareness Questionnaire PSAQ, and performance-based assessment. The results of the study indicate that mixed-/left-handed participants had significantly higher SA than strong right-handed participants when measured by performance-based assessment. Also, in the subjective assessment, mixed-/left-handed participants had significantly higher PSAQ performance scores than strong right-handed participants. The findings of this study suggest that advanced driver assistance systems (ADAS), which show improvement in road safety, should adapt the system functionality based on the driver’s degree of handedness when driving under UFTRs.