The authors present here the results of the first detailed theoretical calculations of differential cross sections for free-free transitions which involve electrons at real target-atom scattering resonances in a laser field. The scattering resonances included, which may occur in the initial or final state (or both) of the free-free absorption, are the lowest 1Se, 1Po, 3Se and 3Po Feshbach resonances of hydrogen occurring at incident electron energies just below the n=2 excitation threshold. They evaluate differential cross sections involving initial 1Se and 3Po resonances and radiation at the CO2 laser wavelength (h(cross) omega =0.117 eV), the double resonances between 1Se and 1Po (h(cross) omega =0.630 eV) and between 3Po and 3Se (h(cross) omega =0.383 eV), and the processes going from 1Se and 3Po resonances to a final state above the n=2 threshold by means of radiation at the Nd laser wavelength (h(cross) omega =1.17 eV). The latter process has three possible final atomic states (1s, 2s, 2p) and corresponds to the joint electron-phonon excitation of the hydrogen atom. It is found that the presence of the electron scattering resonances can enhance the background free-free absorption in the forward direction by as much as four orders of magnitude, but has a relatively minor effect in the backward direction.