Conventional prestack migration does not preserve local directional information of the seismic waves at the image points. New attempts such as sectored migration of azimuth-limited or common-offset-vector data only concern source-receiver azimuth and offset on the surface, which can be poor representation of subsurface wavepath direction. Moreover, they could result in inaccurate imaging because they do not account for the energy propagation between azimuths or offset-vectors. In the past decade, local angle-domain seismic imaging has been highly advocated to avoid migration artifacts and to improve velocity estimation in complex media. Considering prestack time migration (PSTM) is still widely used in seismic imaging and seismic data preconditioning for amplitude variations with offset or incident-angle (AVO/AVA) analysis, fracture detection, and reservoir characterization, we present an azimuth-preserved local angle-domain Kirchhoff PSTM approach for such purposes. We apply a seismic imaging condition in 3D local angle domain and use extended superposition of impulse responses retaining subsurface angular attributes, which are evaluated through the incident and scattering phase slowness vectors using classical-diffraction moveout equations in isotropic, vertical transversely isotropic (VTI) and azimuthally anisotropic media. Two-dimensional synthetic examples demonstrate what the migrated results look like in local angle domain. A wide-azimuth synthetic example with horizontal transversely isotropy (HTI) proves the necessity of azimuthal migration for reliable imaging and azimuthal analysis when azimuthal anisotropy exists in the overburden. Real data examples show the advantages of imaging in subsurface angle domain for properly focusing and revealing azimuth- and angle-dependent variations of residual moveout and migrated amplitudes.