Is there a comparative form of "well"?
Solution 1:
The Oxford English Dictionary says the following. Historically, the comparative of the adverb “well” was “bet” (the link requires subscription) whereas the comparative of the adjective “good” has been “better”. The use of the word “better” as an adverb appeared around the 13th century, and it superseded “bet” by around 1600.
As Kosmonaut writes in a comment on the question, in modern English, the word “better” is used as both an adjective and an adverb. The Merriam-Webster dictionary agrees on this.