Anytime search algorithms solve optimisation problems by quickly finding a (usually suboptimal) first solution and then finding improved solutions when given additional time. To deliver an initial solution quickly, they are typically greedy with respect to the heuristic cost-to-go estimate h. In this paper, we show that this low-h bias can cause poor performance if the greedy search makes early mistakes. Building on this observation, we present a new anytime approach that restarts the search from the initial state every time a new solution is found. We demonstrate the utility of our method via experiments in PDDL planning as well as other domains, and show that it is particularly useful for problems where the heuristic has systematic errors.