From 2:14 to 4:00 is how I like to prepare my cotton:
BTW, I don't actually scrub it with my fingernails or anything like that... just keep rubbing it in one direction on all sides while at the same time also pausing a few times to peel off anything that doesn't want to cooperate, to make it look and feel more like all the cotton fibers are starting to get aligned somewhat. You'll also notice that, the longer you keep on rubbing, the thinner the piece will become, because you are gradually causing it to be more stretched out some. So, as for those spots where it feels like it's still a little bit thicker in comparison to how it should feel like according to you, rubbing in those spots more numerous times will help with getting the correct thickness all along the entire length, albeit I don't really focus too much on the specific part of it that will not be pulled through the coil nor will be pulled inside it, as I can still always thin that part out later anyway─after the tails are cut.
After I cut the tails, I gently thresh out anything that can be threshed out by not actually pulling on it. I know a lot of people just comb out the tails, like almost raking them instead, but I also know that you can do a far better job than they can... I just know that you can.
Cotton is like a type of natural philosophy, or Zen. The brand named UD (Youde) makes a handy little tool called a Cotton Hook, and that I very highly recommend: