Nonlinear continuum generation from diffraction-limited hot spots in rough silver films exhibits striking narrow-band intensity resonances in excitation wavelength. Time-domain Fourier spectroscopy uncovers how these resonances arise due to the formation of a "plasmon staircase", a discreteness in the fundamental oscillation of the plasmon excitations responsible for generating the white-light continuum. Whereas multiple scattering from discrete antennas can be invoked to explain hot spot formation in random assemblies of isolated particles, hot spots in films of fused nanoparticles are excited by interfering propagating surface plasmons, launched by scattering from individual nanoparticle antennas. For closed films, discrete propagating plasmons interact coherently over distances of tens of microns to pump the hot spot.