Plasmonic field-enhanced high-order harmonic generation is investigated theoretically. The plasmonic field is modeled as a spatially inhomogeneous field with a time-dependent cosine squared pulse envelope. A method for solution of the three-dimensional time-dependent Schrödinger equation for a hydrogen atom in such a field is presented. It was found that the harmonic cutoff can be significantly increased. For larger values of the inhomogeneity parameter and longer pulses additional plateaus with higher cutoffs appear in the harmonic spectra. A semiclassical method, which takes into account nonzero initial coordinate and momentum of the ionized electron, is also developed. Using this method the observed multiplateau structure and cutoff positions are explained.