In this article, a three-phase-lag thermoelastic model is developed by extending the Roychoudhuri [24] model (TPL) considering the Taylor series approximation of the equation of heat conduction. This model is an alternative construction to the three-phase lag model. The new model includes high-order of three-phase-lags in the vector of heat flux, and in the thermal displacement and temperature gradients. Also, the models of thermoelasticity of Lord–Shulman (LS), Green and Naghdi (GN), dual-phase-lag (DPL) and TPL are obtained as special cases. We attempt to study the thermal waves in an infinitely long annular cylinder whose inner and outer surfaces are traction free and exposed to known surrounding temperatures in the contexts of this new model. The Laplace transform method is applied to get the solution of the considered problem. We provide a detailed analysis of the results we have obtained. We analyze the results in different tables and graphs and compare them with several models of thermoelasticity. The author believes that the analysis of this study will be useful and important to recognize the basic features of this new model of heat conduction.