The sharp reduction of human mobility in March 2020, as observed by anonymized cellphone data, has played an important role in thwarting a runaway COVID-19 pandemic. As the world is reopening, the risks of new flare-ups are rising. We report a data-driven approach, grounded in strong correlation between mobility and growth in COVID-19 cases two weeks later, to establish a spatial-temporal model of "critical mobility" maps that separate relatively safe mobility levels from dangerous ones. The normalized difference between the current and critical mobility has predictive power for case trajectories during the "opening-up" phases. For instance, actual mobility has risen above critical mobility in many southern US counties by the end of May, foreshadowing the latest virus resurgence. Encouragingly, critical mobility has been rising throughout the USA, likely due to face mask-wearing and social distancing measures. However, critical mobility is still well below pre-COVID mobility levels in most of the country suggesting continued mobility-reduction is still necessary.