Sidelobe raising, caused by multipath and multiple-access interference (MAI), hinders the signal detection and time-delay estimation of multicarrier code-division multiple-access (MC-CDMA) signals, which are crucial for localization and tracking applications as well as the sampling-instant configuration of demodulators. Therefore, we propose a method based on reiterative minimum mean-square error (RMMSE) filters to mitigate the MAI and multipath influence. The RMMSE filter is modified to adapt to the multicarrier modulation and symbol involvement characteristics of MC-CDMA signals. The superiority of the proposed method in terms of sidelobes suppression, compared with existing matched filter, least square estimator, least mean square, and single-carrier RMMSE methods, is verified through simulations in the presence of MAIs and short-time-delay multipath. The results show the advantages of the proposed method in terms of high temporal resolution and robustness to MAI influence. This study could contribute toward multipath detection and aid multipath mitigation techniques in strong MAI scenarios.