Mosquito-borne viruses have been estimated to cause over 100 million cases of human disease annually. Many methodologies have been developed to help identify areas most at risk from transmission of these viruses. However, generally, these methodologies focus predominantly on the effects of climate on either the vectors or the pathogens they spread, and do not consider the dynamic interaction between the optimal conditions for both vector and virus. Here, we use a new approach that considers the complex interplay between the optimal temperature for virus transmission, and the optimal climate for the mosquito vectors. Using published geolocated data we identified temperature and rainfall ranges in which a number of mosquito vectors have been observed to co-occur with West Nile virus, dengue virus or chikungunya virus. We then investigated whether the optimal climate for co-occurrence of vector and virus varies between "warmer" and "cooler" adapted vectors for the same virus. We found that different mosquito vectors co-occur with the same virus at different temperatures, despite significant overlap in vector temperature ranges. Specifically, we found that co-occurrence correlates with the optimal climatic conditions for the respective vector; cooler-adapted mosquitoes tend to co-occur with the same virus in cooler conditions than their warmer-adapted counterparts. We conclude that mosquitoes appear to be most able to transmit virus in the mosquitoes' optimal climate range, and hypothesise that this may be due to proportionally over-extended vector longevity, and other increased fitness attributes, within this optimal range. These results suggest that the threat posed by vector-competent mosquito species indigenous to temperate regions may have been underestimated, whilst the threat arising from invasive tropical vectors moving to cooler temperate regions may be overestimated.