leave "Rescale Islands" enabled to keep hair density consistent.
in the Extended Tools tab of the tool shelf, enable "Use Active.
in Object mode, select the UV map you want your mesh to deform into.
select your edge seams (select only one and then click on Select > Select Similar > Seam).
Now, to deform your mesh into its UV map, you need the UV Shape add-on (not yet updated for 2.8). Hopefully this amazing algorithm will get implemented in an official release some day. You can use it by either installing the original standalone program (not the best because it requires to triangulate your mesh and split the seam edges) or better, by building the Blender SLIM branch (much easier to play with but might be harder to install depending on your OS). For that, I strongly recommend avoiding Blender's built-in UV unwrapping (which introduces important distortions) and instead, using the SLIM algorithm.
Prefer sculpting (combing) your hair manually instead of using the Clump property.
it renders noticeable seams on the islands borders.
it doesn't perform well on long or clumped hair.
I've used this technique on one model so far and it worked well. The idea is to deform your mesh into its UV map while keeping the hair attached to it, so you can render it into a UV texture : I have a technique which might be unorthodox but very efficient.