An unconditionally stable fundamental locally one-dimensional (LOD) finite-difference time-domain (FDTD) method with second-order temporal accuracy and complying divergence (CD) (denoted as LOD2-CD-FDTD) is presented for three-dimensional (3-D) Maxwell's equations. While the conventional LOD-FDTD method does not have complying divergence, the LOD2-CD-FDTD method has complying divergence in a manner analogous to the conventional explicit FDTD method. The update procedures for a family of LOD-FDTD methods that employ similar splitting matrix operators are presented. By extending the previous concept of achieving second-order temporal accuracy for the LOD2-FDTD method via implicit output processing, we hereby propose novel, explicit output processing that not only retains second-order temporal accuracy, but also complying divergence for the LOD2-CD-FDTD method. The current source implementation for the LOD2-CD-FDTD method involves source-incorporation in only the first procedure. To further enhance efficiency, the LOD2-CD-FDTD method is formulated into the fundamental LOD2-CD-FDTD method with efficient matrix-operator-free right-hand sides. Subsequently, detailed implementation for the fundamental LOD2-CD-FDTD method is presented. Analytical proof is provided to ascertain the second-order temporal accuracy of the LOD2-CD-FDTD method. Numerical results and examples are also presented to validate the divergence-complying property of the LOD2-CD-FDTD method.