The resonantly enhanced near-field of micrometer-sized gold antennas has been probed with Angstrom-scale resolution. In situ surface-enhanced infrared spectroscopic vibrational signals of carbon monoxide (CO) layers cold-condensed on the antennas in ultrahigh-vacuum conditions are compared to the signals of CO layers with corresponding thicknesses on a flat gold surface. Vibrational signals of CO as well as the shift of the plasmonic resonance frequency were used to analyze the distance dependence of the near-field. The signal enhancement induced by the antennas not only decays monotonically from the surface but, in contrast to classical near-field models, shows a maximum between 10 and 15 Å and decays also toward the surface of the antenna. This effect is attributed to the spill-out of the electron wave function, as expected from quantum mechanical calculations.