Both 又 and 再 are adverbs meaning "again."
Generally, you place the adverb right before the entire verbal phrase in Chinese, so subject + adverb + verb + object.
Think of "把它关上了" as the entire verbal phrase. You add 又 or 再 before the phrase, to get "又／再把它关上了."
However, for whatever reason, in the second sentence you cited, 再 comes after 会。
In other words, while it is 又会不高兴了, you would definitely say 会再不高兴了.