  Motivated by the observation of partial magnetic order in kagome-based
magnets, we study the classical kagome Ising antiferromagnet, known as kagome
spin ice, including further-neighbor interactions at zero and finite
temperature. While the nearest-neighbor model displays an extensive
ground-state degeneracy, various symmetry-breaking states can appear upon
including additional couplings. Among these, peculiar partially ordered states
have been proposed. We present results from large-scale Monte-Carlo
simulations, establishing that such partial order is stabilized by
third-neighbor couplings along the hexagon diagonals. We show that these states
arise due to the emergence of one-dimensional chains in an entirely frustrated
environment, and that they are stable with respect to further couplings. We
discuss their finite-temperature properties in detail, highlight the magnetic
states' peculiarities depending on the diagonal couplings' sign, and suggest
observing these states using thermodynamic probes.
