Context: Jets are dynamic, impulsive, well-collimated plasma events that develop at many different scales and in different layers of the solar atmosphere. Aims: Jets are believed to be induced by magnetic reconnection, a process central to many astrophysical phenomena. Within the solar atmosphere, jet-like events develop in many different environments, e.g., in the vicinity of active regions, as well as in coronal holes, and at various scales, from small photospheric spicules to large coronal jets. In all these events, signatures of helical structure and/or twisting/rotating motions are regularly observed. We aim to establish that a single model can generally reproduce the observed properties of these jet-like events. Methods: Using our state-of-the-art numerical solver ARMS, we present a parametric study of a numerical tridimensional magnetohydrodynamic (MHD) model of solar jet-like events. Within the MHD paradigm, we study the impact of varying the atmospheric plasma β on the generation and properties of solar-like jets. Results: The parametric study validates our model of jets for plasma β ranging from 10 -3 to 1, typical of the different layers and magnetic environments of the solar atmosphere. Our model of jets can robustly explain the generation of helical solar jet-like events at various β ≤ 1. We introduces the new result that the plasma β modifies the morphology of the helical jet, explaining the different observed shapes of jets at different scales and in different layers of the solar atmosphere. Conclusions: Our results enable us to understand the energisation, triggering, and driving processes of jet-like events. Our model enables us to make predictions of the impulsiveness and energetics of jets as determined by the surrounding environment, as well as the morphological properties of the resulting jets.