Using a template for mapping would be very helpful. I'd post one here but I'm too lazy and on phone, so Google "Head Texture Template", you'll find a 1024x512 one. That's what I recommend to get better detail.

Work on your overall mapping, eye shape, and hair mapping. The best way to improve on that is to watch some anime style art tutorials in shading and blending colors - Shading, Lighting, etc... And how to create the anime eye look. They're all unique so find some you like most and recreate it WITHOUT cheating (cropping, copy and paste)