This is the final setting in the Texture Editor.
Now the UV's are less distorted. But even with this new UV set, I experienced some distortions with the texture. For my next character, I will cut even more parts than what you see in the above picture.

The key to a good texture editor work is not trying to keep all your polygons attached. You should avoid distortion as much as possible.
Cut parts of the mesh where the colors looks solid. Cut parts of your mesh where a seam is undeniably visible.

It is generally a good idea to cut along a mesh joint. Indeed, as the skin of the character folds, the texture's seam will be hidden inside the folding.
If a joint is often wide open and is never hidden inside a folding, you will need to cheat in Photoshop so that the seam does not show any texture roughness. This texture's seam will be sourrounded by plain color. On the edges of the seam, this plain color will gradually be mixed into the asperities of the texture.
If this effect shows up in the final texturing, do not worry, it will never be noticeable in the final animation, while the character moves, gets motion blur, and so on.