Performance of the commonly-used transient hot wire method for measuring the thermal conductivity of fusible materials near solid-liquid phase change is reported with the focus placed on explaining the observed "anomalous" increase of thermal conductivity in relation to the solid-solid transition. Utilizing a 1-D transient formulation and considering a one-step thermal conductivity model, the improved computational methodology captured the monotonic dependence of the predicted thermal conductivity value on the initial temperature of the solid medium more effectively. Hypothesizing that the reported measurements of increase of the thermal conductivity near the solid-liquid phase change temperature are linked to the solid-solid transition, a two-step thermal conductivity model was adopted. This model featured a higher thermal conductivity over a narrow temperature range before a sharp drop upon melting. The predicted values of the thermal conductivity in relation to the initial solid-state temperature were discussed. A rising trend for the predicted thermal conductivity values was observed followed by a smooth decline once the initial solid-state temperature was increased. This predicted trend based on the piecewise thermal conductivity vs. temperature model closely resembled the reported "anomalous" thermal conductivity measurements observed in experimental studies utilizing transient techniques. (C) 2018 Elsevier Ltd. All rights reserved.