Anytime search algorithms solve optimisation problems by quickly finding a usually suboptimal solution and then finding improved solutions when given additional time. To deliver a solution quickly, they are typically greedy with respect to the heuristic cost-to-go estimate h. In this paper, we first show that this low-h bias can cause poor performance if the heuristic is inaccurate. Building on this observation, we then 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. We show that it is particularly useful for hard optimisation problems like planning where heuristics may be quite inaccurate and inadmissible, and where the greedy solution makes early mistakes.