The gradient thermoelasticity problem for a composite rod based on the applied one-parameter model is investigated. To find the Cauchy stresses, the Vishik-Lyusternik asymptotic approach is used, taking into account the presence of boundary-layer solutions in the vicinity of the rods' boundaries and interface. A new dimensionless parameter equal to the ratio of the second rod length and the gradient parameter are introduced. Simplified formulas are constructed in order to find the distribution of the Cauchy stresses depending on the new parameter. After finding the Cauchy stresses distribution, moment stresses, total stresses, displacements, and deformations are further calculated. The dependence of the Cauchy stress jump on the ratio of the rods' physical characteristics and the scale parameter is investigated. The analysis of the results provided is performed.