Freeform reversible embedding of suspended hydrogel (FRESH) is an additive manufacturing technique enabling the 3D printing of soft materials with low or no yield stress. The printed material is embedded during the process until its solidification. From the literature, FRESH abilities are self-healing, reusability, suspending, thermal stability, and high-precision printing. This study proposes a new support hydrogel bath formulation for FRESH 3D printing. To do so, a poloxamer micellar thermoreversible hydrogel is tuned through the addition of poly(ethylene glycol) (PEG) to adapt rheological properties. PEG macromolecules interact with poly(ethylene oxide) blocks of poloxamer and favor micelle dehydration, and then decreasing the gelation temperature, the yield stress, and the viscosity. Parameters such as the Oldroyd number and the Rayleigh-Plateau instability, both dependent on yield stress, were studied to determine their impact on the FRESH 3D printing resolution and accuracy. It was found that print accuracy of embedded parts increases with increasing yield stress but then the self-healing property gets limited, leading to crevasse formation. The usefulness of this approach is distinctly demonstrated through a six-axis printing of a highly complex silicone anatomical model. Printing fidelity of 96.0 ± 3.58% (5-40 mm printed parts) is thus achieved using the newly formulated FRESH material, while only 56.0 ± 0.76% fidelity is obtained using the standard formulation. The present study thus showed that complex FRESH 3D printing of soft materials is possible in this tunable hydrogel and that parts can be manufactured on an industrial scale, thanks to the reusability of the support bath.