I've been in the Czech republic for 1 month now, and every day I meet another Kateřina :P
What I understand is that, 'jina' has a value/emotion attached to it. E.g.
I want another (a different) machine (something's wrong with this one) - jiny stroj
I want another (the next) machine (I'm done with this one) - dalsi stroj
Dalsi actually means next, but can be used as another. Jiny means another/different
Think BEER ;-) Please correct me if I got this wrong, but when you have already drunk one and it's about time to get the second one... Either you want to keep going with the same barrel and then use "dalsi"; or you were not quite happy and want to taste a different beer and then use "jiny".