In this paper, we propose a method to replan coverage paths for a robot operating in an environment with initially unknown static obstacles. Existing coverage approaches reduce coverage time by covering along the minimum number of coverage lines (straight-line paths). However, recomputing such paths online can be computationally expensive resulting in robot stoppages that increase coverage time. A naive alternative is greedy detour replanning, i.e., replanning with minimum deviation from the initial path, which is efficient to compute but may result in unnecessary detours. In this work, we propose an anytime coverage replanning approach named OARP-Replan that performs near-optimal replans to an interrupted coverage path within a given time budget. We do this by solving linear relaxations of integer linear programs (ILPs) to identify sections of the interrupted path that can be optimally replanned within the time budget. We validate OARP-Replan in simulation and perform comparisons against a greedy detour replanner and other state-of-the-art coverage planners. We also demonstrate OARP-Replan in experiments using an industrial-level autonomous robot.