The polyester formed by condensation of 4,4’-bis(6-hydroxyhexoxy)biphenyl (BHHBP) and isophthalic acid has been reported to show many interesting and intriguing thermal properties. By optical microscopy, we established that the polymer shows monotropic liquid-crystalline behaviour. Thus, on heating, no mesophases are formed and the crystal melts directly to the isotropic state; on cooling, however, the material transforms from the isotropic melt to a nematic and smectic A mesophase before crystallizing. However, the DSC heating scan of this polymer was very complex, showing an exotherm and two endotherms; further, unlike the phenomenon of cold crystallization, the exotherm occurred between the two endotherms. If there is only a single phase transition (crystal-isotropic) on heating, then the multiple peaks need to be explained. In this work, we have shown that the multiple peaks in the heating scan are solely related to the thermal history of the sample. The nascent polymer and quench-cooled material showed multiple peaks during heating, whereas slowly crystallized polymer showed a single melting endotherm. By quench cooling from the isotropic melt, it was not possible to obtain the material in an amorphous form, because this polyester had a propensity for fast ordering. In the previous work, on cooling, we only found two instead of three DSC peaks. Here, we have attempted to see if three peaks (corresponding to I-N, N-S-A and S-A-K) can be resolved by going to very slow cooling rates (0.1 degrees C min(-1)). It was found that even at such low cooling rates, I-N and the N-S-A transitions could not be separated into distinct peaks. This is in accord with optical microscope observations which indicated that the transformation to the smectic phase from the nematic was never complete and was overtaken eventually by crystallization. This kind of incomplete phase transformation appears to be a feature of this monotropic liquid-crystalline polymer.