The problem of stabilizing rigid-body attitude dynamics in the presence of pointwise time-delay for the input torque is considered. A quaternion-based linear state feedback controller is shown to achieve local stability in addition to the characterization of sufficient condition that depends only on the magnitude of the initial angular rates. More specifically, no restrictions are imposed on the body initial orientation which is a significant contrast with other results from recent literature that adopt three-dimensional representations for the attitude kinematics. Using the quaternion-based linear feedback structure, the closed-loop system is shown to never admit the possibility for finite-time escapes. While the actual magnitude of the time-delay can be unknown, an upper bound on the delay is assumed to be known. The proof relies on the construction of a functional which does not belong to the family of the strict Lyapunov-Krasovskii functionals, but shares important features with the functionals of this family. The stability conditions and results are illustrated through numerical simulations.