Is this meant like "you're being annoying, don't you have something else to do?" or "have you found employment yet?" because "Do you have any other job?" could be an awkward phrasing depending on the context.
Någon is for en words, något is for ett words, and några is for plural. For example Do you have any other jobs? would be Har du några andra jobb? Since jobb is an ett word it does not change in this case but its plurality is reflected by the adjectives.
Unsure why "jobs" plural isn't accepted, but my grasp of Swedish isn't very good yet so clarification would be nice.
I read it/pictured it as someone showing up to a place to work, being given a task and then asking for a different job for whatever reason.
Asking "Do you have any other job?" sounds really awkward in English, but "Do you have any other jobs?" sounds more natural. Even saying "Do you have another job?" (singular) sounds better than the apparent correct translation.
Do you have another job should be fine. As for plural, the adjectives must agree with the noun. In this case it would be Har du några andra jobb.
How come Do you have any other job? and Do you have some other job? are both accepted translations even though the meaning is different?
Because in the first case I'm just curious to know whether the person has another job, whilst in the second one I'm asking because there's something telling me that the person has some other source of income they're not telling me about.
So is the Swedish sentence ambiguous, or is there a way to re-phrase it to better fit either of the meanings?
Yeah the translation isnt what you'd say in English. More like "do you have another (implying different) job?"
Tinycards has "annat" as meaning "something other." I've been dying to see it used in a sentence, and now I have! So, does annat always (or usually) need the support of något to make sense?
Annat means other. Tinycards is wrong. Något annat means something else. It doesn't need the support of något. It depends on what you're trying to say.
Jag vill ha ett annat öl. - I want another beer.
Jag vill ha något annat. - I want something else.
Your first example, jag vill ha ett annat öl, would actually mean "I want a different beer". Note also that since you used the neuter gender, it means "beer type" rather than "glass of beer".