I know that も particle can be used in place of は、が、を. And it means "in addition", "also".
What function does it have in this adverb "不運にも"? Should we treat it as a separate word here?
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『中上級を教える人のための日本語文法ハンドブック』 explains the expression
不運にも on p. 382 as a sentence adverb (文副詞).
Some adjectives like
卑怯な, etc. can take
も after their conjuntive form to add some evaluation, criticism, or commentary of speaker to the whole sentence.
For example, the sentense
意外にも、彼は集会に現れた。 (Surprisingly, he showed up to the assembly.)
expresses indirectly that the speaker did not expect that he would show up. This unexpectedness is not the main topic of this sentence, but a kind of commentary from the speaker's point of view.
Here is another example:
その警察官は勇敢にもギャングから人質を救出した。 (The policeman bravely rescued the hostages from the gang.)
不運にも could be considered a set phrase. However, technically the も there puts more emphasis on the phrase (強調を加える).
(EDIT: After reading the referenced book mentioned by Tsutomu it can be concluded that this usage is fixed for certain words. I suggest everybody to read the linked page from google books.)
Perhaps other examples would make it more clear:
も in the above sentence emphasizes just how bad the debt is getting.
Another example of
にも is when you want to emphasize どうしてもできない:
歩くにも歩けないほど人が多い (There are so many people I can't even walk)
This will always be in the form ～にも～ない or ～にも～ず.
And yet another example is when lining things up (列挙)
日本にもアメリカにもメリットがあります (There are benefits for both America and Japan)
There are perhaps other examples of the usage also not mentioned above.