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Parker Solar Probe's (PSP) discovery of the prevalence of switchbacks (SBs), localised magnetic deflections in the nascent solar wind, has sparked interest in uncovering their origins. A prominent theory suggests these SBs originate in the lower corona through magnetic reconnection processes, closely linked to solar jet phenomena. Jets are impulsive events, observed across scales and solar atmosphere layers, associated with the release of magnetic twist and helicity. This study examines whether self-consistent jets can form and propagate into the super-Alfv\'enic wind, assesses the impact of distinct Parker solar wind profiles on jet dynamics, and determines if jet-induced magnetic untwisting waves display signatures typical of SBs. We employed parametric 3D numerical MHD simulations using the ARMS code to model the self-consistent generation of solar jets. Our study focuses on the propagation of solar jets in distinct atmospheric plasma $\beta$ and Alfv\'en velocity profiles, including a Parker solar wind. Our findings show that self-consistent coronal jets can form and propagate into the super-Alfv\'enic wind. Notable structures such as the leading Alfv\'enic wave and trailing dense-jet region were consistently observed across diverse plasma $\beta$ atmospheres. The jet propagation dynamics are significantly influenced by atmospheric variations, with changes in Alfv\'en velocity profiles affecting the group velocity and propagation ratio of the leading and trailing structures. U-loops, prevalent at jet onset, do not persist in the low-$\beta$ corona, but magnetic untwisting waves associated with jets show SB-like signatures. However, full-reversal SBs were not observed. These findings may explain the absence of full reversal SBs in the sub-Alfv\'enic wind and illustrate the propagation of magnetic deflections through jet-like events, shedding light on possible SB formation processes., Comment: 24 pages, 14 figures, published in Astronomy & Astrophysics