First of all, having a stiff head will do you no good. None at all. So before you go along animating anything with a figure, it's a really good idea to just twist the head a whole lot. The head will then come to a point where it is slightly more loose, after that you can't really make it any more loose by twisting it. It won't be REALLY lose though, don't worry, only loose enough so that it's easy enough to twist without as much resistance which can cause some big problems.
It should now be much easier to move the head smoothly without messing up a hairpiece.
Unfortunately, though, at this point the hairpiece can still be accidentally moved. It can be especially problematic if your character has a loose helmet that prevents you from touching the face, however, still shows the figures eyes or something. To counter this, simply apply a bit of sticky stuff the top of the figure's head, and replace his/her hair/hat/helmet.
If applied correctly, the stick should allow you to move the head by holding the hair rather than he face, which is very easy. I've done this in most of my films since How to Not Rob a Bank.
The only downside is that changing the hair/hat/helmet later on will take a bit of time as it can take a little while to remove the stick.
If you've done it properly, it shouldn't take long at al the dab the residue out from the hair and head. However, if you've been a bit careless in its application, it could take several minutes of trying to get a little bit if stick from a tiny nook in a knight's helmet.
It's best to apply the stick very lightly around the stud on the minifigure's had. Try to stay away from the stud itself.
For sticky stuff, I use a kneadable eraser, since it doesn't dry out, it's grey, and is easily removed from aa surface by simply dabbing it with some of the stick.
Sticky stuff also has many other applications in brickfilming.
I would avoid using tape, as the residue from tape can be somewhat more difficult to remove.