  Liquid crystals can self-organize into a layered smectic phase. While the
smectic layers are typically straight forming a lamellar pattern in bulk,
external confinement may drastically distort the layers due to the boundary
conditions imposed on the orientational director field. Resolving this
distortion leads to complex structures with topological defects. Here, we
explore the configurations adopted by two-dimensional colloidal smectics made
from nearly hard rod-like particles in complex confinements, characterized by a
button-like structure with two internal boundaries (inclusions): a two-holed
disk and a double annulus. The topology of the confinement generates new
structures which we classify in reference to previous work as generalized
laminar and generalized Shubnikov states. To explore these configurations, we
combine particle-resolved experiments on colloidal rods with three
complementary theoretical approaches: Monte-Carlo simulation, first-principles
density functional theory and phenomenological $\mathbf{Q}$-tensor modeling.
This yields a consistent and comprehensive description of the structural
details. In particular, we characterize a nontrivial tilt angle between the
direction of the layers and symmetry axes of the confinement.
