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.