<p>A highly effective Subgradient-based Tableau Pivot (STP) method is developed for sparse signal recovery (compressive sensing). The algorithm utilizes subgradient information of the objective function and is implemented via tableau operations. For the noiseless observation model, the STP method is not only significantly more accurate and faster than typical methods based on minimizing estimation error, such as Basis Pursuit Denoising or Least Absolute Shrinkage and Selection Operator, but also greatly improves computational speed compared to methods based on the Basis Pursuit (BP) optimization framework using standard simplex method at the same accuracy level. A notable advantage of the proposed STP method is that all relevant operations are performed on the original matrix size, rather than doubling the number of columns as required when reformulating the BP framework as a linear program. The robustness of the proposed algorithm in the presence of observation noise is thoroughly analyzed. Simulation results are presented to verify the established theoretical findings and to compare the performance of the proposed algorithm with other state-of-the-art recovery methods reported in the literature. Notably, the proposed algorithm demonstrates outstanding computational efficiency in terms of runtime.</p>