ruudygh as we are not directly simulating the external shape of the pillow (like you would be doing with a cloth solver like n-cloth), this is actually a more tricky request than it might at first appear.
If the animation is far away from origin, and the initial shape of the tissue is different than that of the default model, you could do the following:
Animate the position of the solver over 1 frame (with substeps keyframed to 1 substep over that frame)
Use an art directable rest shape and keyframe it on over a small number of frames (where the target if the rest shape is the desired "initial state")
You will still need some run-up if you do this, but you should be able to dramatically reduce the length of the run-up.