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In this paper we consider the rearrangeable multi-plane banyan-type switching fabrics, called also log2(N, 0, p) switching networks, with even number of stages. For such networks different rearranging algorithms have been proposed for both: one-at-a-time and simultaneous connection models. In this paper we consider the one-at-a-time connection model, where connections arrive to the system one-by-one, and in case of blocking rearrangements are realized. To our knowledge, known algorithms require several rearrangements, and the number of such rearrangements have not been considered in the literature. We propose the new rearranging algorithm for the multi-plane banyan-type switching fabric composed of even number of stages. This algorithm leads to success using only one rearrangement. We also introduce the modified version of this new algorithm, in which rearrangement of an existing connecting path can be realized without its interruption.