You can use "another" before a noun whenever you can use "a (n)" before a noun. The rules are the same. Another = an other! other + plural or uncountable nonspecific noun

Another implies extra or additional with respect to the current amount. "I'm going back for another sandwich." Other is an adjective implying it is different than the object in discussion.

A simple rule to help you remember the difference between another and other is: another + singular noun other + plural noun others (a pronoun to replace other + plural noun) I need another cup. (cup is singular so we use another) I need other cups. (cups is plural so we use other) I need others. (refers to other cups)

The primary difference between another and other is that both another and other are determiners, but another precedes singular countable nouns or pronouns. Conversely, other precedes a plural countable as well as uncountable nouns and pronouns.