Prior works on near-field beam training have mostly assumed dedicated polar-domain codebook and on-grid range estimation, which, however, may suffer long training overhead and degraded estimation accuracy. To address these issues, we propose in this paper new and efficient beam training schemes with off-grid range estimation by using conventional discrete Fourier transform (DFT) codebook. Specifically, we first analyze the received beam pattern at the user when far-field beamforming vectors are used for beam scanning, and show an interesting result that this beam pattern contains useful user angle and range information. Then, we propose two efficient schemes to jointly estimate the user angle and range with the DFT codebook. The first scheme estimates the user angle based on a defined angular support and resolves the user range by leveraging an approximated angular support width, while the second scheme estimates the user range by minimizing a power ratio mean square error (MSE) to improve the range estimation accuracy. Finally, numerical simulations show that our proposed schemes greatly reduce the near-field beam training overhead and improve the range estimation accuracy as compared to various benchmark schemes.