In the sentence:
If 初めて is an adverb, it should modify the phrase behind it, which is a rule of Japanese. So here it should modify「その本の本当の価値が分かるようになった」, but why there is a comma "、" between them?
After all, I don't see elsewhere an adverb and its modifying part are seperated by a comma. So I feel the original sentence is a little strange.
I came up with an explanation: maybe here we should see て初めて as a compound particle, which means "not until". But I think it's also Ok to see 初めて as a single adverb, and the sentence should be
just like the sentence:
in the second explanation of 初めて from デジタル大辞泉