Vibration data often carries important information about the dynamic characteristics of structures, but continuous collection of high-frequency vibration data can create practical challenges (e.g., transmission and storage of large amounts of data). Compressive sensing (CS), which integrates data acquisition and compression utilizing the sparse nature of signals, shows promise in mitigating the practical challenges of data measurement, transmission, and storage. The presence of side lobes in the discrete Fourier transform (DFT) of vibration signals, known as the spectral leakage effect, adversely impacts the sparsity of these signals. This, in turn, diminishes the effectiveness of CS algorithms. Built with three key modifications, this study proposes a spectral compressive sampling matching pursuit (CoSaMP) algorithm to alleviate the effect of spectral leakage. First, the redundant DFT dictionary is incorporated to consider the spectral leakage effect. Second, a coherence-inhibiting large component identification step replaces the original one, addressing the coherence issues due to the redundant DFT dictionary. Third, Ridge regression supersedes the least squares in the estimation step to improve the reconstruction accuracy. The proposed spectral CoSaMP algorithm is validated by the simulated data from a frame structure and the field data collected from a long-span cable-stayed bridge. The influence of related parameters (i.e., frequency redundancy indicator, coherence threshold, sparsity level, and regularization parameter of Ridge regression) is extensively investigated, and recommended parameter settings are provided. Moreover, the superiority of spectral CoSaMP over classic CS algorithms (including classic CoSaMP, ℓ 1 -norm regularized optimization, and Bayesian CS) on the reconstruction accuracy of vibration signals is demonstrated through comparative studies.