The pandemic of COVID-19 is undoubtedly one of the biggest challenges for modern healthcare. From the beginning of the outbreak, new variants have also proven to be even more contagious, accelerating the spread and stressing the healthcare system. In order to analyze the spatio-temporal aspects of the spread of COVID-19, technology has helped us to track, identify and store information regarding positivity and hospitalization, across different levels of municipal entities. In this work, we present a method for predicting positive and hospitalized cases at the department level of France via a novel multi-scale graph neural network, integrating information from fine-scale geographical zones of a few thousand inhabitants. Our model manages to outperform baselines and deep learning models, presenting low errors in both prediction tasks. We specifically point out the importance of our contribution in predicting hospitalization since hospitals became critical infrastructure during the pandemic. To the best of our knowledge, this is the first work to exploit high-resolution spatio-temporal data in a multi-scale manner, incorporating additional knowledge, such as vaccination rates and population mobility data. We believe that our method may improve future estimations of positivity and hospitalization, which is crucial for healthcare planning.