In this paper we complement and improve fixed point results established on TAC-contractive mappings in the framework of $b$-metric spaces. Our analysis starts by a recent paper of Babu and Dula [G. V. R. Babu and T. M. Dula, Fixed points of generalized TAC-contractive mappings in $b$-metric spaces, Matematički Vesnik 69 (2017), no. 2, 75--88].