You’ll get the best results if your UV layout is flat like in your first picture and not bent like in the second one. You can quickly do that with the Textools addon. One way to remove the waviness is by baking a fairly round barrel first and removing edge loops with Ctrl+X after baking.
But this can break the normalmap, because the original mesh normals do not match the baked ones anymore. The last barrel in my picture has this problem on the top.