(Here it is, as a reminder: לא תלכנה בנות ישראל פרועות ראש בשוק, אחת פנויה ואחת אשת איש
The daughters of Israel should not walk with uncovered heads in the marketplace, both unmarried and married women)
While I was looking through the Tur to find it, I found that earlier in the same siman, there's another relevant snippet of halakha- "אסור לשמוע קול ערוה או לראות שערה" It is forbidden to listen to the voice of a sexually-forbidden-woman, or to look at her hair. The same thing is brought in the Shulhan Arukh, again word-for-word (E"H 21:1).
This is presented in the context of a variety of prohibitions on male attention to women, much of which I am, at the least, quite uncomfortable with- as you can probably tell by the beginning of the sentence. Nevertheless, it presents an interesting balance to the following se'if, which somehow changes the feeling of the thing, for me. I shouldn't show it, and they shouldn't look. If I fall through, they're still not allowed to enjoy my error.
Somehow, I manage not to react to this text by putting my head-covering into that category of "do this to protect your too-easily-tempted men from sin", a philosophical approach to modesty that often irks me. I don't know why it doesn't press those buttons, since it would be logical enough. The chain of reasoning would be: 1. they oughtn't look. Therefore 2. I should have to keep it out of sight. I dislike the one-sided responsibility for someone else's obligation that goes on in this approach. Expressing the requirement for women to cover their heads as a separate requirement helps prevent this from being my dominant reading of the rule. The other meanings that head-covering has both for me and in the text, also keeps it from falling into this hole for me.