Thanks for the file, KévinH,
The core problem is that you like to have two qualities in one. Having no Stretchiness at all, and then have full.
Please have a look here:
I introduced a lot of Stretchiness and then limited it via a Vertex Tag. This is a quick one; you might explore it more. For this, I have created a Dummy figure to keep the calculation time short. Such a dummy can be set invisible and work for the Simulation while the actual Figure is visible.
As you can see, when you come to the last frame, large stretched polygons take over the head. That could be refined in the Vertex Map, but I left it as is, as it showcases the problem. Polygons can be pulled on, but the local "resolution" lowers, and SDS or Cloth-Surface will not fully remedy this. There are limits, or you can increase the mesh to the point that it will get dense.
As so often, I would use the Simulation to get to a certain point and animate from there.
All the best