Confining light to sharp metal tips has become a versatile technique to study optical and electronic properties far below the diffraction limit. Particularly near-field microscopy in the mid-infrared spectral range has found a variety of applications in probing nanostructures and their dynamics. Yet, the ongoing quest for ultimately high spatial resolution down to the single-nanometer regime and quantitative three-dimensional nanotomography depends vitally on a precise knowledge of the spatial distribution of the near fields emerging from the probe. Here, we perform finite element simulations of a tip with realistic geometry oscillating above a dielectric sample. By introducing a novel Fourier demodulation analysis of the electric field at each point in space, we reliably quantify the distribution of the near fields above and within the sample. Besides inferring the lateral field extension, which can be smaller than the tip radius of curvature, we also quantify the probing volume within the sample. Finally, we visualize the scattering process into the far field at a given demodulation order, for the first time, and shed light onto the nanoscale distribution of the near fields, and its evolution as the tip-sample distance is varied. Our work represents a crucial step in understanding and tailoring the spatial distribution of evanescent fields in optical nanoscopy.