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The modeling of the near-field interaction in the scattering-type scanning near-field optical microscope (s-SNOM) is rapidly advancing, although an accurate yet versatile modeling framework that can be easily adapted to various complex situations is still lacking. In this work, we propose a time-efficient numerical scheme in the quasi-electrostatic limit to capture the tip-sample interaction in the near field. This method considers an extended tip geometry, which is a significant advantage compared to the previously reported method based on the point-dipole approximation. Using this formalism, we investigate, among others, nontrivial questions such as uniaxial and biaxial anisotropy in the near-field interaction, the relationship between various experimental parameters (e.g. tip radius, tapping amplitude, etc.), and the tip-dependent spatial resolution. The demonstrated method further sheds light on the understanding of the contrast mechanism in s-SNOM imaging and spectroscopy, while also representing a valuable platform for future quantitative analysis of the experimental observations.