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This paper studies a direct-sequence spread-spectrum communication system with a small spreading factor (e.g., a single digit) in underwater acoustic multipath channels. Exploiting the channel characteristics that the propagation paths can be grouped into distinct clusters, a receiver with a set of parallel branches is proposed, where per-survivor processing (PSP) is applied on each branch to deal with the signal from one cluster while explicitly treating the signals from other clusters as structured interference. As such, it overcomes a major limitation of an existing PSP based receiver [Zhou, Morozov, and Preisig, J. Acoust. Soc. Am. 133, 2746-2754 (2013)], avoiding exponential complexity increase when the spreading factor decreases. On the first branch, joint channel estimation and interference cancellation are performed based solely on the survivor paths. On the other branches, joint channel estimation and interference cancellation are carried out based on the survivor paths and the tentatively decoded data. The bit log-likelihood ratios from different branches are combined for channel decoding. Performance results based on simulations and collected data sets validate the superior performance of the proposed receiver over the conventional RAKE receiver, which is effective only when the spreading factor is large.