Unwrapping complex cylinder to save on texture space

Support Materials and Textures
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1

So I have this cylindrical shape

Screenshot_1
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After unwrapping, every vertical face is created as big “circular uv-face” that’s wasting a lot of uv-space. I’m trying to turn them into straight belts but then, the effect is like this

Screenshot_2
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Texture then becomes stretched. I’m aware that it’s due to the differences in lenghts between edges, but even if I straighten the uv-island, and try to manually scale down it’s upper part, it does not fix the problem.

So my question is, if it’s even possible to straighten uv-islands of a cylinder, that are oriented vertically (going inward, idk how to professionally call it). And here’s the view on my actuall uv-map. You can see, why I want these circles to turn into belts.

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2

If you want strips, add seam across and for each ring. Select quad near seam, reset UV, L with seam enabled, and follow active quad.
For rings, add logical seams. L with seam enabled, and UV project from view. Instead of wasting so much space, I’d loose some texel information and store each ring inside eachother.