In this paper, we propose new algorithms for evacuation problems defined on dynamic flow networks. A dynamic flow network is a directed graph in which source nodes are given supplies (i.e., the number of evacuees) and a single sink node is given a demand (i.e., the maximum number of acceptable evacuees). The evacuation problem seeks a dynamic flow that sends all supplies from sources to the sink such that its demand is satisfied in the minimum feasible time horizon. For this problem, the current best algorithms are developed by Schlöter (2018) and Kamiyama (2019), which run in strongly polynomial time but with highorder polynomial time complexity because they use submodular function minimization as a subroutine. In this paper, we propose new algorithms that do not explicitly execute submodular function minimization, and we prove that they are faster than those by Schlöter (2018) and Kamiyama (2019) when an input network is restricted such that the sink has a small in-degree and every edge has the same capacity.