Elastic full-waveform inversion (FWI) can provide accurate and high-resolution subsurface parameters. However, its high computational cost prevents the application of this method to large-scale field-data scenarios. To mitigate this limitation, we propose a target-oriented elastic FWI methodology based on a redatuming step that relies upon an extended least-squared migration process. In our approach, the surface-reflection data can be attributed to a given subsurface portion when mapped into the image space. This process allows us to reconstruct reflection data generated by a target area and recorded with a virtual acquisition geometry positioned directly above it. The redatuming step enables the application of an elastic FWI method within the target portion only. The entire workflow drastically decreases the overall cost of the surface-data inversion and allows the retrieval of accurate elastic parameters of the area of interest. We demonstrate the effectiveness of our approach on a synthetic case based on the well-known Marmousi2 model and on a 3D ocean-bottom-node (OBN) pressure data recorded in the Gulf of Mexico in which we retrieve the elastic parameters of a potential prospect, positioned in proximity of a salt-dome flank, and whose rock-physical properties are consistent with the presence of a gas-bearing sand reservoir.