We report on the pressure dependence of the Hall coefficient RH in quasi-2D heavy fermion CeCoIn5. At ambient pressure, below a temperature associated with the emergence of non-Fermi liquid properties, RH is anomalously enhanced. We found that the restoration of the Fermi liquid state with applied pressure leads to a gradual suppression of this dramatic enhancement. Moreover, the enhancement in RH was found to be confined to an intermediate temperature window, where inelastic electron-electron scattering is dominant. Our results strongly support the presence of cold and hot spots on the Fermi surface probably due to anisotropic scattering by antiferromagnetic fluctuations, which may also prove relevant for the debate on the anomalous normal-state properties of high-T-c cuprates.