Inspired by the investigation of Kalb–Ramond strings and hairy black holes by Duncan et al., in this paper we investigate two approaches to determine the existence of asymptotic flatness and how the presence of cosmic string axionic torsion hair affects wormholes and black holes. In the case of Einstein–Cartan–Kalb–Ramond (ECKR) gravity, it is shown that by assuming asymptotic flatness in ECKR equations, the KR vanishes at both sides far away from the wormhole throat, indicating the absence of torsion hair. On the other hand, in the case of teleparallel gravity, we show that in a Schwarzschild black hole with a cosmic string, by considering the cosmic string tension as constant, the torsion invariant vanishes, showing no signatures of torsion hair. When a dynamical cosmic string with variable string tension is introduced in torsional wormholes, it is shown that no asymptotic flatness is obtained, and torsion does not vanish asymptotically. This indicates that dynamical cosmic strings must generate an imprint of torsion hair at wormholes away from the throat. We argue that in the future, the ideas here may be used to detect torsion from an astrophysical point of view.