This hat encourages good behaviour: improving posts. This is exactly what a good badge should do.
Why not make it a permanent bronze badge? There is an existing badge to encourage people to edit at all, but this one would encourage users to put in enough effort that the post is actually improved.
It should only be awardable once. Making it awardable multiple times would lead to some annoying gaming (such as sniping edits on posts you recognize as good) and would need higher requirements.
There is a year-old badge request for editing a negatively-scored post which later gets a positive score, but I think the hat's criteria is a better choice. The negatively-scored posts that need the most help are probably below -2, and even with a good edit you'll need quite a bit of luck to bring it back over 0.
There are also a lot of very meh-quality posts with a score around 1–3 (particularly given the new review queue) which a good edit could improve enough to gain four more votes, but this behaviour would not be incentivised by the existing proposal. Requiring a relative change in score makes the tag somewhat less luck-based and encourages the positive behaviour in more situations.
(The badge also the unique attribute that it is already in use in a different form. We can look at how often the hat was awarded for an idea of how often the badge would be, and to some extent (despite the hat being "secret") we see if people try to game it too badly.)
Thoughts? Name suggestions?