Three high-spin phases recently discovered in the spin-crossover system Mn(taa) are identified through analysis by a combination of first-principles calculations and Monte Carlo simulation as a low-temperature Jahn-Teller ordered (solid) phase, an intermediate-temperature dynamically correlated (liquid) phase, and an uncorrelated (gas) phase. In particular, the Jahn-Teller liquid phase arises from competition between mixing with low-spin impurities, which drive the disorder, and intermolecular strain interactions. The latter are a key factor in both the spin-crossover phase transition and the magnetoelectric coupling. Jahn-Teller liquids may exist in other spin-crossover materials and materials that have multiple equivalent Jahn-Teller axes.