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We numerically construct asymptotically global AdS_{3}×S^{3}×T^{4} black holes in type IIB supergravity, with R_{t}×SO(2)×SO(3)×U(1)^{4} symmetry, localized on the S^{3} and translationally invariant along the torus. These solutions, with horizons whose spatial cross sections have S^{4}×T^{4} topology, dominate the microcanonical ensemble at low energies. At higher energies, a first-order phase transition occurs to BTZ×S^{3}×T^{4} black holes-possessing R_{t}×SO(2)×SO(4)×U(1)^{4} symmetry and S^{1}×S^{3}×T^{4} horizon topology. By the AdS/CFT correspondence, this transition reflects the spontaneous breaking of the SO(4) R symmetry of the D1-D5 CFT_{2} to SO(3). We also compute the expectation value of the scalar operator with lowest conformal dimension in the low-energy phase. Our SO(3)-localized black holes-together with the U(1)^{2}-localized solutions of Bena et al. [J. High Energy Phys. 10 (2025) 165.JHEPFG1029-847910.1007/JHEP10(2025)165]-point to a rich landscape of novel black holes that may approach the CFT_{2} "sparseness bootstrap condition," and shed light on how macroscopic entanglement in thermal phases encodes microscopic structure via internal directions.