Owing to their remarkable thermoelastic and scale-dependent physical properties, carbon nanotubes (CNTs) have emerged as promising reinforcements to enhance the thermomechanical response of nano-tailored composite materials. Two fundamental aspects influencing the thermoelastic response of a nano-tailored composite are investigated herein; namely, CNT waviness and the existence of an interphase between a CNT and the polymer matrix. We propose a systematic micromechanical modelling scheme in conjunction with a new interphase model to determine the coefficients of thermal expansion (CTEs) of a nano-tailored composite. The proposed modelling approach has been applied to a case study involving the need to determine the effective CTEs of a novel nano-tailored composite–fuzzy carbon fiber heat exchanger. The results reveal that (i) the interphase between a CNT and the surrounding polymer matrix plays a crucial role in the modelling of the thermoelastic properties of the CNT-based composite, (ii) planar orientation of CNT waviness has a significant influence on the effective CTEs of the hybrid nano-tailored composite, and (iii) for the particular planar orientation of CNT waviness and the value of CNT wave frequency, the effective CTEs of the hybrid nano-tailored composite become zero, making the nanocomposite a “super-insulator”.