The frequency range of an electron cyclotron current drive is explored, in order to enhance the current drive efficiency. Its model is based on a two-dimensional, relativistic, bounce-averaged Fokker-Planck equation. The current drive efficiency is enhanced by a largely Doppler-shifted (up-shifted) frequency, using the relationship Y<or approximately= gamma -1, where Y=l omega ce/ omega (l:harmonic number, omega ce: electron cyclotron frequency, omega : injection frequency) and gamma is the relativistic factor. The behavior of the enhanced efficiency is examined under the toroidal effect as well as the DC electric field effect.