Whilst studying I came across this,
From Sefer Kedushah, Issurei Biah Chapter 21 - Halacha 17
Jewish women should not walk in the marketplace with uncovered hair. [This applies to] both unmarried* and married women.
Followed by this note,
*I.e., a widow or a divorcee. A woman who never married may wear her hair uncovered (Chelkat Mechokek 21:2).
Simply put, may under any circumstances a never married/divorced woman cover her hair? In the note it says "may wear her hair uncovered", which implies that such a woman could cover her hair and need never to have been married or be currently married to do so. It sounds like it is permissible.
I also wonder if it is allowed for a unmarried woman for personal tznius reasons to cover her hair.
When speaking of "personal tznius reasons" this discussion came up in a class. As some married women choose to, for instance, never uncover their hair as it is seen to be meritorious. The question was asked if it were allowed for unmarried women to have a similar, but of course situational different, high standard.
Consulting my local Orthodox rabbi is a given. However, I like hearing different perspectives. Sometimes LORs don't elaborate on their answers or answer so simply that a discussion doesn't benefit from the answer.