Accurate direction-of-arrival estimation with hybrid analog--digital millimeter-wave sensor arrays is important for localization, environment sensing, and measurement beam control for sensing applications. However, the limited number of radio-frequency chains and training beams in practical hardware makes it difficult to approach the angular resolution of fully digital arrays. This paper develops a covariance-guided discrete Fourier transform (DFT) beam selection framework tailored to beamspace ESPRIT for hybrid millimeter-wave receivers. A short hybrid training phase realizes a virtual centro-symmetric subarray and yields a sample covariance that is processed by forward--backward averaging, nonnegative least-squares power and noise fitting, and a Toeplitz positive-semidefinite projection to reconstruct a denoised full-aperture covariance matrix. This covariance is then used to score and select, within each coarse sector, small contiguous blocks of DFT beams that concentrate signal energy and preserve effective aperture under a strict beam budget. The selected beams feed a sparse beamspace ESPRIT stage that operates only on actually available adjacent beam pairs, so that the overall complexity is dominated by a single low-dimensional ESPRIT call. Monte Carlo simulations for a thirty-two-element uniform linear array with three paths indicate that, in the considered scenarios, the proposed method can reduce the gap to the Cramér--Rao bound, lower the failure rate, and provide favorable accuracy--runtime trade-offs compared with a sectorization-based baseline built from the same codebook and estimator. For the unitary DFT codebook studied here, the fine-stage beam selector reduces to a covariance-guided contiguous energy-window rule; the broader score formulation also accommodates non-unitary effective beam dictionaries arising from hardware non-idealities.