Solving \emph{stochastic games} with the reachability objective is a fundamental problem, especially in quantitative verification and synthesis. For this purpose, \emph{bounded value iteration (BVI)} attracts attention as an efficient iterative method. However, BVI's performance is often impeded by costly \emph{end component (EC) computation} that is needed to ensure convergence. Our contribution is a novel BVI algorithm that conducts, in addition to local propagation by the Bellman update that is typical of BVI, \emph{global} propagation of upper bounds that is not hindered by ECs. To conduct global propagation in a computationally tractable manner, we construct a weighted graph and solve the \emph{widest path problem} in it. Our experiments show the algorithm's performance advantage over the previous BVI algorithms that rely on EC computation.