First of all, what are you doing? "Headshot" is a term borrowed for modelling/ theatrical agency lingo pertaining to a head and shoulder or tight closeup portrait used for publicity or modelling or acting /entertainment portfolio or what used to be called a "glossy. Some folks use the term "headshot" for a business or corporate portrait, etc. Some cleit or the agencies want an authentic unretouched or embellished image, others want a more flattering image, and she may need a port study- a dramatic image that is extremely sharp and revealing- kind wats and all.

I can understand if you want a "sharper" lens than the one you are presently using but having a faster lens, as you specified, does not necessarily provide more sharpness. Do you want a faster lens, one with a wider aperture, to provide better "bokeh", more selective focus, more shallow depth of field? If you have an image where the eyes are tack sharp and the ears and background are significantly softer, the perception of sharpness is more pronounced.

I use my Canon 24-105 for portraiture so that I can change focal lengths repeatedly during sessions to accommodate closeup, 3/4 length and full-length images without changing lenses. I have prom lenses that are sharper, however, that lens at f/11 is sharp enough - 2 two attached images were made with the lens at 105mm.
These are relatively low-resolution files/ The 16x20 prints made for the original files were too sharp! Most of my clients do not was to see their pores in their "headshots"!

If you need a better rendition of texture- that is a function of lighting.