We extend a recently developed dynamically self-consistent model of the MilkyWay constrained by observations from the Gaia observatory to include a radiallyanisotropic component in the dark matter (DM) halo, which represents the debrisfrom the accreted Gaia-Sausage-Enceladus (GSE) galaxy. In the new model, whichwe call a self-consistent Anisotropic Halo Model or scAHM, we derivedistribution functions for DM velocity in heliocentric and geocentric referenceframes. We compare them with the velocity distributions in the standard halomodel (SHM) and another anisotropic model (SHM++). We compute predictedscattering rates in direct-detection experiments, for different target nucleiand DM particle masses. Seasonal dependencies of scattering rates are analyzed,revealing small but interesting variations in detection rates for differenttarget nuclei and DM masses. Our findings show that the velocity distributionof the anisotropic GSE component significantly deviates from Gaussian, showinga modest impact on the detection rates. The peculiar kinematic signature of theradially anisotropic component would be most clearly observable bydirection-sensitive detectors.