Fact 1: As we all (ought to) know, 'another' is nothing more than the simple combination of an indefinite pronoun (an) and an adjective (other) which happens to be written as one word: an + other = another. Fact 2: In a list of two or more nouns, you often only need to say the article once. For example, in the sentence below, the 'a' of 'a blue suit' also covers the following two noun phrases. There is no need to repeat 'a': "He was wearing a blue suit, white shirt and striped tie". In your first example, you have already said 'a' once in 'a school closure', so there is no need to include the 'an' part of 'another'. That would be a needless repetition of the indefinite article. In your second example, you have to include the 'an' of 'another' because the previous noun, 'Advanced Options', was a proper noun (the name of the submenu) and had no article.

Another is basically a contraction of "an other" you need to establish a singular with "submenu". It is one of 2 direct objects in your second sentence and your first "advanced options" is plural.