But "another" can easily be both jiný or další.
The famous line from Mario - "The princess is in another castle." is better translated as "Princezna je v jiném hradě."
"Princezna je v dalším hradě." would correspond better to "The princess is in the next castle."
On the other hand, when you're on a train and look out of the window and say: "Hey, I see another castle!" - that would be "Hele, vidím další hrad!" - in this context, "jiný" would clearly mean "different".