He hurt his right arm, so he used his other arm to bandage himself. It is "other" and not "another". Here are examples with another: http://sentence.yourdictionary.com/another

Replace "another" with "other" and the sentence would be grammatically correct. "another" should be used when there isn't a specific noun to refer to. In your sentence, the person has two arms and one is injured so the we know the other arm exists already. An example: There are too many of us so we will need another car. I hope this helped!