You usually find adding a few more loop cuts will improve distortion. It also depends of your UV’s have been made rectangular using something like UVSquares or TexTools ‘rectify’.
This is because the simple (and so fastest) affine texture mapping is not “correct”… also you are mapping a rectangular sectioned texture to a non retangular surface (the inner rings are smaller than the outer so it is a non- parallelogram)…
But actually not the probelm because you only see this because you are using some rectangualar texture… if you paint (especially in 3D) this thing you proberly would not see any “distortions”.
You might want to use @colkai’s mentioned trick… or unwrap every side of this modified cyclinder seperately and paint it in 3D.