*I don't think I understood your question. What is the reason use such deformed UV? What is your base mesh? And what is your final shape?
Until someone who undertood better answer your Q ... here is a tip.*

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It seems to me easier to select all faces *Unwrap* > **Reset**.  
With all selected and one face selected as *Active* (bright one) *Unwrap* > **Follow Active Quad**

[![enter image description here][1]][1]

Like that you can get even distribution again for your texture.

[![enter image description here][2]][2]


  [1]: https://i.sstatic.net/TiCn6.png
  [2]: https://i.sstatic.net/GZHqz.png