じん is the demonym reading (i.e. the reading in names for people coming from a particular location). It also might be the more common reading in compounds that end in 人 that characterise particular groups of people.
If a compound starts with 人 and has a falling pitch accent (i.e. the accented syllable is the first one – it’s high – and everything else falls from there), then the reading is usually じん
That aside, I think you have to learn things by rote. Aim to remember entire phrases (i.e. ‘this compound is…’) rather than ‘the right reading here is…’. I think that will be helpful.
The answer @Rihn linked to is very interesting, but I think the rules for when it’s the ‘first kanji’ are much more rarely followed. The ‘second kanji’ rules are much more useful.
Some other (more complete) answers to consider, with lots of examples:
These two answers present many more cases, including those that don’t fall into the rules from the answer Rihn mentioned. I hope this helps. The readings usually stick with practice anyway, provided, once again, that you aim to learn the readings of compounds as single units before breaking them down.