Not familiar with this particular hair but what I try to do with most transmapped hairs is;
connect the texure also to bump and to displacement. Play with the settings.

Also the red-brown looks too vivid;
I would put an HSV node between the texture and diffuse_color to reduce the vivid color.
and-or reduce the diffude_value of the hair.

Did you set the texture to crisp? Try none?

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