Because local slant stacking increases the data dimension in beam migration, the volume of local slant stacks can be enormous and can obstruct efficient data processing. In addition, a proper beam compression algorithm can reduce the computation of ray tracing and beam mapping. Thus, compressing the local slant stacks with high fidelity can improve the efficiency of beam migration. A new approach is proposed to efficiently compress the local slant stacks. This approach combines the estimation of multiple local slopes based on the structure tensor to reduce the number of slopes, and the sparse representation for the slant stacked data via the matching pursuit decomposition to reduce the number of temporal samples. Furthermore, a new algorithm to estimate multiple local slopes based on the second-order structure tensor is proposed to handle the intersecting events efficiently. Several data examples indicated that the new compression algorithm required much less storage. Meanwhile, the new algorithm can restore the significant events and tolerate some random noise. The migration results determined that this compression algorithm does not obviously degrade the quality of the beam migration result, and it even makes the migration result more clear by suppressing the random noise smearing.