CUtting out hair

hey there, i’ve asked this on reddit but the subreddit where i asked this at hardly ever gives me answers and my friend wants to get this done so i can get to working on other models,

(not sure if pictures really work from this link so ill just repost it

within this image, the outline from being selected isn’t exactly on the hair but generally around it, which is a problem because it transfers over to being black/white spots, or in my case the same color as the hair, i cant think of any solution that’d just cut the transparent spots for me since thats the only thing i can think of to resolve the issue.