The Voynich Ninja

If the hat were a later addition, it would make our job a bit easier, but I'm almost certain it isn't. The linework looks quite integrated with the rest of the drawing.

About clothing terminology: if we know the proper names it might be easier to find help in the literature. For example, now we know the hat is one of two options:

• a chaperone without the usual frills, so it would just be the "bourrelet" part
  
• a completely different hat type that looks like a bourrelet but did not evolve from a hood like the chaperone did
  

Apparently there are a two main names for this kind of padded roll: bourrelet and roundlet. It doesn't matter much whether this roll has a piece of fabric covering the crown of the head or not.

What we see in the VM Gemini man is a specific moment in the evolution of the garment. In the beginning of the 15th century people were tired of draping their hood+cape over their heads each time to look fashionable, so the headgear was redesigned into a padded roll for ease of use. The sleeve and rooster comb were then sewn to this roll. What we are looking for (for Gemini) is the period where it was fashionable to wear the bourrelet/roundlet without these attachments. 

Some preliminary reading I've done suggests that the surge in popularity of this particular style will coincide nicely with the VM carbon daterange. Most clear cut examples are about 1400-1440. But more work needs to be done on this.

I think you're going to find that these fashions developed in the south (Italian provinces) and spread North. So the exact date will depend upon where you assume the manuscript was drawn.