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Hyperspectral tomography in THz amplitude contrast is presented using a time-domain spectroscopy system in the spectral range 0.3-2.5 THz. The Fourier transformed signal data is used to reconstruct test objects' cross-sections at multiple frequencies using standard filtered backprojection. The full hyperspectral set of images reconstructed at around 300 adjacent spectral points is used to trace the combined contribution of Beer-Lambert volume attenuation, Fresnel reflection losses and Rayleigh roughness scattering losses, which is in good overall agreement with the experimental results. The image quality for Styrofoam (refractive index around 1.02, attenuation coefficient < 1 mm-1) test objects is best in the range 0.8-2.0 THz depending on the porosity of the material.