좀 더 = 조금 더 = a little more (O)
많이 더 = a lot more (O)
아니 더 = no more (X)
“No more ~,” or “There isn’t ~ anymore,” in Korean is 더이상 ~ 이 없다 / 더이상 ~하지 않다. So, “There is no milk anymore,” would be 더이상 우유가 없어요. (FYI, in Korean, I wish we could simply say 더이상 우유 or something simple for no more milk like you do in English, but 더이상 always accompanies a negative verb).
좀 더 먹다 = eat a little more (O) You can also say “좀 더 많이”
많이 더 빠른 = a lot more fast = much faster (Makes sense) You can say “더 많이 빠른”
아니 더 우유 = no more milk (X)
i.e.
Please walk a little bit more: 좀 더 걸으세요.
The amount which is a little bit more: 좀 더 많은 양 (To make “a little bit more” adjective, you add 많은 after 좀 더).
A lot more fast train: 더 많이 빠른 열차/기차
This trains goes a lot more fast: 이 기차가 더 많이 빨리 가요.
Hope this helps. Let me know if you have any questions.