Suboptimal search algorithms can often solve much larger problems than optimal search algorithms, and thus have broad practical use. In the past decade several algorithms have been proposed that improve performance over the basic weighted A*, but these algorithms rely special heuristics and data structure implementations, meaning that they require significantly more effort to implement and use than basic weighted A*. The goal of this paper is to provide a simple algorithm that is easy to implement and can achieve state-of-the-art performance. The paper does this in three ways. First, it studies the influence of node re-openings during search, showing the potential savings and cost of re-openings. As most existing suboptimal algorithms re-open states, turning off re-openings can often improve performance. Second, it studies termination conditions, providing a better termination condition for approaches like Optimistic search. Finally, taken together with recently-developed priority functions, a general framework for Improved Optimistic Search is developed that is both simpler than existing algorithms and also has better performance in practice.