Wrinkling represents one of the failure modes in sandwich structures, although in practical designs the loss of strength and global buckling often occur at lower compressive loads. However, the properties of both polymeric matrix in the facings as well as the polymeric core degrade under an elevated temperature. As a consequence, wrinkling that does not present a problem at the room temperature may become a dominant mode of failure at elevated temperatures. In this article, we suggest that a reinforcement of the core material with stiff random nanoscale or microscale reinforcements may alleviate wrinkling. The solution accounts for the thermal loading history and the effect of temperature on the stiffness of the materials of the core and facings. While nano or microscale reinforcements increase the capacity of the structure to resist wrinkling, the strength of the core may be compromised due to the presence of such inclusions in the core material. Accordingly, the residual stresses in the reinforced core are evaluated using a finite-element method and accounting for the effect of temperature on the properties and stresses. It is demonstrated that both wrinkling and the core strength analyses should account for the effect of temperature on the material properties.