We consider surveillance applications in which sensors are deployed in large numbers to improve coverage fidelity. Previous research has studied how to select active sensor covers (subsets of nodes that cover the field) to efficiently exploit the redundant node deployment. Little attention was given to studying the tradeoff between fault tolerance and energy efficiency in sensor coverage. In this work, our objective is to rapidly restore coverage of the field under unexpected node failures. For this purpose, we explore different adaptable coverage strategies. We then propose design guidelines for applications employing distributed cover-selection algorithms to control the degree of redundancy at local regions in the field and achieve energy-efficient coverage. In addition, we develop a new distributed technique to facilitate switching between active covers without the need for node synchronization. Distributed cover selection protocols can be integrated into our framework (referred to as "resilient online coverage" (ROC)). We evaluate the effectiveness of ROC through analysis and extensive simulations.