Fare thee well,
I am certain that the verse about women's dress code would be more detailed if this factor was that essential. If there was explicitly supposed to be a cover for the head the Quran would have mentioned the word for head and it does not as far as I know. And as we know God said that His book is very thorough on what is important for us to know.
As I said in a previous post, the verse about clothing is in conjunction with the verse which says that women should not strike their heels to attractively shake their bodies and with traditional muslim women clothing this is not possible since nothing will be revealed anyway. That is why I believe the whole "Islamic" clothing to be a hoax and more about ancient traditions.
A woman, as far as my understanding goes, should not provoke 'unnecessary' sexual arousal and cover private parts. This is what I think Quran wants to convey in terms of information.
The almost same command was issued for men too and there is not much difference really. Women seems to have been more directly targeted because of an innate desire to be attractive and this slight difference is present in Quran.
The whole hijab thing is an invention as much else in the man-made religion Islam. God only told people to have a decent appearance and no hocus pocus.
However, more clothes can undoubtedly cause a person to seem more modest and provoke less attraction, but God hardly meant for women to dress like ghosts to barely be recognized. Lengthen the garments was told in Quran and I suppose this is the whole purpose of the verses.
I get a feeling that where the Quran was written, the women used to dress in very revealing ways and act quite immodestly and it was to get the community's women to dress more decently that Quran pronounced that garments should be lengthened. This was later abused and became more and more absurd to include so called Burqa, Niqab and (Iranian) Chador.
So from to have been called to stop the women from being inappropriately immodest the clergymen made it into extremism with oppressive dress codes instead. So from have been uttered by God to tell women to stop being too loose it was over-interpreted to suppress women completely and say women's mere beauty is immoral and sinful.
Humans have a tendency to always exaggerate everything that they do and it is also true for religion. It is dangerous when humans start with conjecture. Reason is strangely not allowed in religion. When it comes to spirituality humans are very mentally unstable.
God bless you