Jiný (m.) / jiná (f.) / jiné (n). means "different"
Další (same in all genders) means "next" or "additional"
The English word "other"/"another" translates either to "jiný" or "další" depending on whether its meaning is closer to "different" or "additional/next".
There are many tourists here. Look, there's another one! (DALŠÍ)
I don't like this table, I need to buy another one. (JINÝ)
"Jiný" is a hard adjective. Hard adjectives always end in "-ý" in their dictionary (i.e. singular masculine) form. It changes forms as follows:
singular: jiný muž (m.), jiná žena (f.), jiné pivo (n.)
plural: jiní muži (m. anim.), jiné stromy (m. inanim.), jiné ženy (f.), jiná piva (n.)
As you can see, "jiní" is one of the possible forms: masculine animate plural. This doesn't make it a soft adjective.
An example of a soft adjective is "cizí" - foreign. It ends in "-í" in the dictionary form as well as in all nominative forms. In the nominative, it doesn't change its form based on gender or number: cizí muž, cizí žena, cizí pivo, cizí muži, cizí stromy, cizí ženy, cizí piva.