OK, since we are (some anyway) speculating. I once heard (cant remember where) that the reason for women covering their heads went back to Genesis chapter 6 concerning the sons of God making marriage and babies who became giants with human women. The head covering meant to hide their beauty from any randy angels who may want to relive the old Genesis 6 days before the flood. Don't know if there is any truth to that. Has anyone here ever heard of this?
To be honest, I don't know when it became a thing, but I know that Jewish women covering their hair is alluded to at the Trial of the Suspected Adultress, when the Cohen unbinds the woman's hair, as part of the trial of humiliation (in the hopes that she would either confess or simply give up).
On a positive note, there is a passage from Solomon's "Song of Songs" that list many physical attributes of a woman. Since it was mentioned, it became part of Jewish law (or, perhaps Jewish law was already there, and the hint from Song of Songs just highlighted how) Jewish women should cover up certain things so that she can use them to excite her husband. These things include her hair (although this only pertains to married women, for some reason), her upper arms, her thighs, her breasts, and interestingly enough, her singing voice. (The last one never sat well with me, as I love to sing and am good at it.)
But again, it's not to HIDE her, and it isn't about SHAME. It's about acknowledging her beauty, and praising her modesty.