חלקת מחוקק: This is one source I can't transliterate

The Helkat Mehokek, R. Moshe Lima, (the link is to a Hebrew Wikipedia article, since there doesn't seem to be one in English yet), was born in 1604 in Poland, and died in 1658.  He served as a rabbi in Slonim, Vilna and Brisk.  Here's what he has to say about our favorite piece of Shulhan Arukh (i.e. Even HaEzer 21:2)

אחת פנויה ואחת אשת איש: פנויה בעולה קאמר, אבל בתולה אמרינן דיוצאה בהינומא וראשה פרועה, וכן הוא בב"ח ועיין במרדכי

Both an unmarried and a married woman (the citation from Shulhan Arukh): this means an unmarried woman who has had sex, but we say that a virgin may go out with only a hinuma and an uncovered head, and thus in the Bach, and see the Mordecai.  

What a hinuma is, I'm still working on.  According to Jastrow (p. 348, right column), it is either a curtained litter on which a virgin bride is carried or a slumbering couch.  This seems unlikely to be any sort of reality in 17th century Poland, so he must mean something else.  According to my pocket dictionary, it's a bridal veil.  Neither seem like regular clothing for anyone...  Even-Shoshan has the same two definitions, although expands the latter to any sort of face veil, as well.  Still, that doesn't seem to make a lot of sense with what I think I know of 1600s Poland- I don't think face-veiling was normative there.  I'm wondering if it isn't some sort of babushka for the outdoors, worn over loose hair.

Checking in the Arukh HaShalem sent me to the place where the Talmud discusses exactly this question (I seem to be in extremely good company): Ketubot 17b.  The two options given are, 1. a dome of myrtle branches, and 2. a veil she can doze in.  Rashi suggests that this second interpretation involves a scarf over her head that can hang down and cover her eyes, and says about it "as they do in our place".  I don't know if that's talking about young women in general, or about brides... (Also, Rashi lived several centuries before R. Lima, and in a different country.)

What I can tell is that head-covering for unmarried women is at the least less stringent than for married women, but there is some item that seems to be required for public places.  Any further insight into the identity of the hinuma in the view of Ashkenazi Acharonim (or anyone else, for that matter) would be welcomed...