English can use definite and indefinite articles before other thing (or other used as a noun) in the same way it uses them before nouns in general; by contrast, Spanish, for example, does not use an indefinite article before otra cosa. But instead of writing an other thing, English writes another thing, losing a space.
So it is "You need to buy another book" in the same way it is "You need to buy a red book" developing from "You need to buy a book", but "You need to read my other book" coming from "You need to read my book".
You can break another apart as in "I'm not a partner or a bit-on-the-side, I'm a significant other".