In the present paper, the nonlinear transfinite element procedure recently published by the author is improved by introducing an enhanced convergence criterion to significantly reduce the computational run-times. It is known that transformation techniques have been developed mainly for linear systems, only. Due to using a huge number of time steps, employing the conventional time integration methods requires quite huge computational time and leads to remarkable error accumulation, numerical instability, or numerical damping, especially for long investigation times. The present method specially may be extended to problems where the required time steps are of the order of the round-off errors (e.g., coupled thermoelasticty problems). The present procedure is employed for transient thermoelastic analysis of thick-walled functionally graded cylinders with temperature-dependent material properties, as an example. To reduce the effect of the artificial local heat and stress shock source generation at the mutual boundaries of the elements, second order elements are used. Influences of various parameters on the temperature and stress distributions are investigated. Furthermore, results of the proposed transfinite element technique are compared with the results obtained by other references to verify the validity, accuracy, and efficiency of the proposed method.