  In $\alpha$-RuCl$_3$, an external magnetic field applied within the honeycomb
plane can induce a transition from a magnetically ordered state to a disordered
state that is potentially related to the Kitaev quantum spin liquid. In zero
field, single crystals with minimal stacking faults display a low-temperature
state with in-plane zigzag antiferromagnetic order and a three-layer
periodicity in the direction perpendicular to the honeycomb planes. Here, we
present angle-dependent magnetization, ac susceptibility, and thermal transport
data that demonstrate the presence of an additional intermediate-field ordered
state at fields below the transition to the disordered phase. Neutron
diffraction results show that the magnetic structure in this phase is
characterized by a six-layer periodicity in the direction perpendicular to the
honeycomb planes. Theoretically, the intermediate ordered phase can be
accounted for by including spin-anisotropic couplings between the layers in a
three-dimensional spin model. Together, this demonstrates the importance of
interlayer exchange interactions in $\alpha$-RuCl$_3$.
