Terahertz (THz) ultra-massive multiple-input multiple-output (UM-MIMO) systems are promising in enabling next-generation wireless communications, offering high data rates with tens of GHz of continuous bandwidth and high spectral efficiency. In THz UM-MIMO systems, a new paradigm of cross-field communications is emerging, since THz transmission distances span from near-field to far-field. To achieve the benefits of THz UM-MIMO, precise beam alignment implemented through beam training or beam scanning is required. However, different from the traditional far-field alignment in the angle domain, the near-field angle and distance alignment should be considered in the cross-field. The additional distance domain searching brings higher training overhead and thus limits the system's performance. In this paper, a far-field training with estimation (FTE) framework for cross-field beam alignment is proposed. The far-field training enables the received signal-to-noise ratio (SNR) in both the far- and near-field for successful control signal reception. Moreover, a three-phase beam estimator (TPBE) is proposed for high-precision alignment. Extensive simulations demonstrate the effectiveness of the proposed methods. Specifically, the FTE possesses a near-optimal signal-to-noise ratio with only 0.5 dB deviation, with 3.3% training overhead and low complexity compared to near-field exhaustive search.