Elastic full-waveform inversion (FWI) can generally yield subsurface model parameters of high spatial resolution. However, the computational and storage burden is tremendous. To mitigate this problem, we developed the following three strategies: First, we derived a time-space-domain adaptive staggered-grid finite-difference method. The size of the calculation was reduced by using smaller operator lengths for high S-wave velocities. Second, we evaluated a multiscale elastic FWI scheme based on the second-generation wavelet, by which scale decompositions of data and model space were implemented. With the scale increasing, spatial sampling intervals and time steps became larger. Therefore, the minimum computational cost could be achieved by combining the variable-operator-length scheme with the multiscale scheme. Third, to economize the memory consumption, we extended the efficient boundary storage and checkpointing schemes into elastic FWI. Only wavefields of several parts in the boundary area between two adjacent checkpoints were stored and used to reconstruct source wavefields of the inner area. Furthermore, the optimal number of checkpoints could be automatically determined by minimizing the amount of storage. We validated these new strategies for the SEG/EAGE overthrust model. The synthetic example suggested the feasibility and robustness of the new elastic FWI method in terms of improving computational efficiency and alleviating large amounts of data storage. We also analyzed the accuracy of our FWI method. The inversion results revealed that our wavelet-based method had better reconstructed structures and higher convergence rate than the conventional filtering-based multiscale FWI method.