In this work, we study a six-dimensional movable antenna (6DMA)-enhanced Terahertz (THz) network that supports a large number of users with a few antennas by controlling the three-dimensional (3D) positions and 3D rotations of antenna surfaces/subarrays at the base station (BS). However, the short wavelength of THz signals combined with a large 6DMA movement range extends the near-field region. As a result, a user can be in the far-field region relative to the antennas on one 6DMA surface, while simultaneously residing in the near-field region relative to other 6DMA surfaces. Moreover, 6DMA THz channel estimation suffers from increased computational complexity and pilot overhead due to uneven power distribution across the large number of candidate position-rotation pairs, as well as the limited number of radio frequency (RF) chains in THz bands. To address these issues, we propose an efficient hybrid-field generalized 6DMA THz channel model, which accounts for planar wave propagation within individual 6DMA surfaces and spherical waves among different 6DMA surfaces. Furthermore, we propose a low-overhead channel estimation algorithm that leverages directional sparsity to construct a complete channel map for all potential antenna position-rotation pairs. Numerical results show that the proposed hybrid-field channel model achieves a sum rate close to that of the ground-truth near-field channel model and confirm that the channel estimation method yields accurate results with low complexity.