Hair strands clinging to the old background almost always means the selection was too coarse. Channel cutout uses the brightness difference between color channels to build the selection, then pulls black and white apart with Levels and patches the edge with the Brush. Understand the principle and you stop wrecking the result.
The core idea, in plain terms
A channel is really just a record of color information. We use the brightness difference between the subject and the background inside one particular channel to generate a selection. Put simply: make the part you want pure white and the part you don't want pure black. Once that principle clicks, everything after it is just fiddling.
Channel cutout in five steps
Where one stroke won't come clean, patch it with a black-and-white brush at around 80% hardness. Small gray noise specks: hit them with white or black.
If the subject went transparent instead of the background, click the mask thumbnail and press Ctrl+I to invert it.
A few rants and tips for beginners
First, channel cutout suffers most from over-compressed source files. Once a JPG has collapsed into mosaic blocks, nobody can rescue it. Use RAW or a high-quality image whenever you can.
Second, don't lock yourself to one channel. Sometimes red is hopeless and blue is perfect. I spent my early days glued to the green channel before realizing that switching is far more efficient.
Third, if the hair edge still looks washed out, the Levels pull wasn't clean enough. Go over it with the layer mask and the Brush, or refine it with Refine Edge inside Select and Mask.
Anyway, once channel cutout is in your fingers, you'll feel prepared for semi-transparent objects and complicated edges. If you don't have PS installed, the online version I mentioned earlier at https://www.zaixianps.cc is easy to practice on. Run it on a few difficult images and you'll find the whole thing is actually pretty mindless.