Select and mask
Select and Mask is a separate workspace in Photoshop for refining selections, especially tricky edges like hair and fur. With the Refine Edge brush you fish loose strands out of the background. It produces clean, natural cutouts.
Start from a rough selection and open Select and Mask via the options bar. Use the Refine Edge brush along hair boundaries to make the transition smooth, and play with Shift Edge and Feather. Choose a layer mask as output, so everything stays editable. Tip: set the view to On Layers or On Black to see which stray edges are still hanging on.
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The presets on this site set these adjustments up for you as a starting point, which you then fine-tune to taste.