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A little help with もう...?

Dec 20, 2014
I want to be sure I'm understanding this correctly. So, you have もう which can mean two general ideas.

The first is "already" or "no longer" depending on the verb.
Example: もうスーパーに行った。 = I already went to the super market.
Example: 姉はもう肉を食べない。 = My sister no longer eats meat.

The second is, I guess, "another" or "other" at least that is what I got from all the reading I do. However, I can't seem to find anything online that directly tells me this, so I'm still not 100% sure.
Example: さあ、もう一回! = Now, one more time!
Example: コーヒーもう三杯をお願いします。 = Another three cups of coffee please.

Is my hypothesis correct? Any explanations would be much appreciated. o3o