Massive multiple-input multiple-output (MIMO) for 5G is evolving into the extremely large-scale antenna array (ELAA) to increase the spectrum efficiency by orders of magnitude for 6G communications. ELAA introduces spherical-wave-based near-field communications, where channel capacity can be significantly improved for single-user and multi-user scenarios. Unfortunately, the near-field region at large incidence/emergence angles is greatly reduced with the widely studied uniform linear array (ULA). Thus, many randomly distributed users may fail to benefit from near-field communications. In this paper, we leverage the rotational symmetry of uniform circular array (UCA) to provide uniform and enlarged near-field regions at all angles, enabling more users to benefit from near-field communications. Specifically, by exploiting the geometrical relationship between UCA and users, the near-field beamforming technique for UCA is developed. Based on the analysis of near-field beamforming, we reveal that UCA is able to provide a larger near-field region than ULA in terms of the effective Rayleigh distance. Moreover, a concentric-ring codebook is designed to realize efficient codebook-based beamforming in the near-field region. In addition, we find out that UCA could generate orthogonal near-field beams along the same direction when the focal point of the near-field beam is exactly the zeros of other beams, which has the potential to further improve spectrum efficiency in multi-user communications compared with ULA. Simulation results are provided to verify the effectiveness of theoretical analysis and feasibility of UCA to enable more users to benefit from near-field communications by broadening the near-field region.