"Set edge flow" for UVs?

Hi folks,

Does anyone know if there is a way to “set edge flow” for UV edges?

I have to conform a UV map to an existing boundary from a reference image (for print work) and I am finding it quite hard to keep my lines parallel along curves. Well, without lots of manual editing.

Or perhaps there is a tool I am overlooking? I thought “minimize stretch” might help it, but the iterative approach seems to make the UVs look a total mess.