Context. 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 phenomena, observed at various scales in different solar atmosphere layers, associated with the release of magnetic twist and helicity. Aims. This study examines whether self-consistent jets can form and propagate into the super-Alfvénic wind, assesses the impact of different Parker solar wind profiles on jet dynamics, and determines if jet-induced magnetic untwisting waves display signatures typical of SBs. Methods. We employed parametric 3D numerical magnetohydrodynamics (MHD) simulations using the Adaptively Refined Magnetohydrodynamics Solver (ARMS) code to model the self-consistent generation of solar jets. Our study focuses on the propagation of solar jets in distinct atmospheric plasma β and Alfvén velocity profiles, including a Parker solar wind. We explored the influence of different atmospheric properties thanks to analysis techniques such as radius-time diagrams and synthetic in situ velocity and magnetic field measurements, akin to those observed by PSP or Solar Orbiter. Results. Our findings demonstrate that self-consistent coronal jets can form and then propagate into the super-Alfvénic wind. Notable structures such as the leading Alfvénic wave and trailing dense-jet region were consistently observed across different plasma β atmospheres. The jet propagation dynamics are significantly influenced by atmospheric variations, with changes in Alfvén velocity profiles affecting the group velocity and propagation ratio of the leading and trailing structures. U-loops, which are prevalent at jet onset, do not persist in the low-β corona but magnetic untwisting waves associated with jets exhibit SB-like signatures. However, full-reversal SBs were not observed. Conclusions. These findings may explain the absence of full reversal SBs in the sub-Alfvénic wind and illustrate the propagation of magnetic deflections through jet-like events, shedding light on possible SB formation processes.