This paper considers a millimeter wave (mmWave) integrated sensing and communication (ISAC) system, where a base station (BS) equipped with a large number of antennas but a small number of radio-frequency (RF) chains emits pencillike narrow beams for persistent tracking of multiple moving targets. Under this model, the tracking lost issue arising from the misalignment between the pencil-like beams and the true target positions is inevitable, especially when the trajectories of the targets are complex, and the conventional Kalman filter-based scheme does not work well. To deal with this issue, we propose a Transformer-based mmWave multi-target tracking framework, namely m3TrackFormer, with a novel re-acquisition mechanism, such that even if the echo signals from some targets are too weak to extract sensing information, we are able to re-acquire their locations quickly with small beam sweeping overhead. Specifically, the proposed framework can operate in two modes of normal tracking and target re-acquisition during the tracking procedure, depending on whether the tracking lost occurs. When all targets are hit by the swept beams, the framework works in the Normal Tracking Mode (N-Mode) with a Transformer encoder-based Normal Tracking Network (N-Net) to accurately estimate the positions of these targets and predict the swept beams in the next time block. While the tracking lost happens, the framework will switch to the Re-Acquisition Mode (R-Mode) with a Transformer decoder-based Re-Acquisition Network (RNet) to adjust the beam sweeping strategy for getting back the lost targets and maintaining the tracking of the remaining targets. Thanks to the ability of global trajectory feature extraction, the m3TrackFormer can achieve high beam prediction accuracy and quickly re-acquire the lost targets, compared with other tracking methods.