The Qur'an does not order women to cover their hair, face, or any part of the head. There is one main verse in the Qur'an which refers to women's dress code, that is verse 24:31. The verse outlines that women should be modest and keep their chest area / cleavage covered in public:
Tell the believing women to lower their gaze and guard their chastity, that they should not display their charms / attractions beyond what is normally apparent, and that they should draw their clothes over their chests / cleavage. They should not reveal their charms / attractions except to their husbands, or their fathers, or fathers of their husbands, or their sons, or the sons of their husbands, or their brothers, or the sons of their brothers, or the sons of their sisters, or their women, or those committed to them by oath, or the male servants who are without need, or the child who has not yet understood the composition of women. Let them not strike with their feet in a manner which reveals their charms / attractions. Turn to God in repentance, oh you who believe, so that you may be successful. (24:31) وَقُلْ لِلْمُؤْمِنَاتِ يَغْضُضْنَ مِنْ أَبْصَارِهِنَّ وَيَحْفَظْنَ فُرُوجَهُنَّ وَلا يُبْدِينَ زِينَتَهُنَّ إِلا مَا ظَهَرَ مِنْهَا وَلْيَضْرِبْنَ بِخُمُرِهِنَّ عَلَى جُيُوبِهِنَّ وَلا يُبْدِينَ زِينَتَهُنَّ إِلا لِبُعُولَتِهِنَّ أَوْ آبَائِهِنَّ أَوْ آبَاءِ بُعُولَتِهِنَّ أَوْ أَبْنَائِهِنَّ أَوْ أَبْنَاءِ بُعُولَتِهِنَّ أَوْ إِخْوَانِهِنَّ أَوْ بَنِي إِخْوَانِهِنَّ أَوْ بَنِي أَخَوَاتِهِنَّ أَوْ نِسَائِهِنَّ أَوْ مَا مَلَكَتْ أَيْمَانُهُنَّ أَوِ التَّابِعِينَ غَيْرِ أُولِي الإِرْبَةِ مِنَ الرِّجَالِ أَوِ الطِّفْلِ الَّذِينَ لَمْ يَظْهَرُوا عَلَى عَوْرَاتِ النِّسَاءِ وَلا يَضْرِبْنَ بِأَرْجُلِهِنَّ لِيُعْلَمَ مَا يُخْفِينَ مِنْ زِينَتِهِنَّ وَتُوبُوا إِلَى اللَّهِ جَمِيعًا أَيُّهَ الْمُؤْمِنُونَ لَعَلَّكُمْ تُفْلِحُونَ
It should be noted that God has left parts of this verse deliberately vague for a good reason. The verse encourages a modest dress code for women and discourages exposing parts of the body which are beyond the normal appearance of a modest woman. The non-specific nature of the word 'charms / attractions' allows this to also be referring to behavioral attractions and not exclusively to women's dress. The verse does, however, specifically mention that the chest / cleavage area should be covered in public. The Arabic expression used here is 'place the KHIMAR over the chest / cleavage'. The word KHIMAR is from the trilateral Arabic root letters Kha-Ma-Ra خ م ر , which literally means any type of 'cover' or 'covering'. Anything that covers can be a KHIMAR, including a curtain, a table-cloth or a blanket, as well as items of clothing. In fact, the Arabic word for 'intoxicants' is KHAMR, which is from the same Arabic root Kha-Ma-Ra خ م ر precisely because it covers or clouds the mind. Hence, the word KHIMAR does not refer specifically to a headscarf. The verse simply states that women should be modest and keep their chest area covered in public.
In verse 33:59, God instructs the Prophet to tell all believing women to draw their clothing over them so that they would not be 'abused / harmed':
Oh prophet, tell your wives, your daughters, and the women of the believers that they should draw their clothing around them. That is better that they will be known and not abused / harmed. God is Most Forgiving, Most Merciful (33:59) يَا أَيُّهَا النَّبِيُّ قُلْ لأَزْوَاجِكَ وَبَنَاتِكَ وَنِسَاءِ الْمُؤْمِنِينَ يُدْنِينَ عَلَيْهِنَّ مِنْ جَلابِيبِهِنَّ ذَلِكَ أَدْنَى أَنْ يُعْرَفْنَ فَلا يُؤْذَيْنَ وَكَانَ اللَّهُ غَفُورًا رَحِيمًا
It is important to consider the context of this verse in order to realise that the verse is not a general guideline on women's dress-code, but is in-fact referring to a specific instruction to the Prophet on how to protect believing women from abuse and persecution which they faced by the hypocrites of that time. The Arabic word used for 'harm / abuse' in this verse is based on the trilateral root letters A-ZA-YA أ ذ ي, which is the same word in the surrounding verses describing the abuse and persecution faced by the believers. This can be seen when we read the verse in context in the Qur'an:
Indeed, those who abuse / harm ( أ ذ ي ) God and His messenger, God has cursed them in this world and in the Hereafter, and has prepared for them a humiliating punishment. (33:57) نَّ الَّذِينَ يُؤْذُونَ اللَّهَ وَرَسُولَهُ لَعَنَهُمُ اللَّهُ فِي الدُّنْيَا وَالآخِرَةِ وَأَعَدَّ لَهُمْ عَذَابًا مُهِينًا
Those who abuse / harm ( أ ذ ي ) believing men and believing women unjustly bear on themselves the crime of slander and great sin. (33:58) وَالَّذِينَ يُؤْذُونَ الْمُؤْمِنِينَ وَالْمُؤْمِنَاتِ بِغَيْرِ مَا اكْتَسَبُوا فَقَدِ احْتَمَلُوا بُهْتَانًا وَإِثْمًا مُبِينًا
Oh prophet, tell your wives, your daughters, and the women of the believers that they should draw on them their clothing. That is better that they will be known and not be abused / harmed ( أ ذ ي ). God is Most Forgiving, Most Merciful (33:59) يَا أَيُّهَا النَّبِيُّ قُلْ لأَزْوَاجِكَ وَبَنَاتِكَ وَنِسَاءِ الْمُؤْمِنِينَ يُدْنِينَ عَلَيْهِنَّ مِنْ جَلابِيبِهِنَّ ذَلِكَ أَدْنَى أَنْ يُعْرَفْنَ فَلا يُؤْذَيْنَ وَكَانَ اللَّهُ غَفُورًا رَحِيمًا
If the hypocrites and those in whose hearts is a disease and those who spread falsehood among people in the city do not stop, We shall let you overpower them and they will not be able to remain your neighbours except a while. (33:60) لَئِنْ لَمْ يَنْتَهِ الْمُنَافِقُونَ وَالَّذِينَ فِي قُلُوبِهِمْ مَرَضٌ وَالْمُرْجِفُونَ فِي الْمَدِينَةِ لَنُغْرِيَنَّكَ بِهِمْ ثُمَّ لا يُجَاوِرُونَكَ فِيهَا إِلا قَلِيلا
Verse 33:59 is therefore not about women's dress code but a specific instruction dealing with the threat of hypocrites' persecution and abuse at the Prophet's time. The context makes clear that the purpose of the instruction was to warn hypocrites not to abuse / harm believing women when they are known to them, otherwise they face the danger of retaliation from the believers.
When discussing dress code for women in general, it is important to keep things in proportion and not overemphasise something which God Himself regards as less pertinent than righteousness and faith in Him. In the following verse, God reminds us that what matters to Him essentially is to have a character of righteousness, and all the 'material' or worldly matters come second:
Oh Children of Adam, We have provided you with clothing to cover yourselves as well as for luxury. However, the best clothing is the clothing of righteousness. This is among the signs of God, so that perhaps they will reflect. (7:26) يَا بَنِي آدَمَ قَدْ أَنْـزَلْنَا عَلَيْكُمْ لِبَاسًا يُوَارِي سَوْآتِكُمْ وَرِيشًا وَلِبَاسُ التَّقْوَى ذَلِكَ خَيْرٌ ذَلِكَ مِنْ آيَاتِ اللَّهِ لَعَلَّهُمْ يَذَّكَّرُونَ
The word HIJAB in Arabic, based on the trilateral root letters Ha-Ja-Ba ح ج ب, literally means a 'screen' or 'barrier', ie. something which separates one thing from another. The word does not mean a 'headscarf' or 'head-covering'. In fact, God makes use of this word eight times in the Qur'an, seven of these is in the noun form HIJAB and one is in the passive form MAHJUBUN. In all of these cases, the meaning is always in the sense of a 'screen', 'partition', or 'barrier'. It is never referring to headscarves, head-covering or any part of a women's dress. Therefore, the word 'Hijab' has nothing to do with women's dress code. The following are all the verses in the Qur'an which make use of Ha-Ja-Ba ح ج ب :
God does not order women to cover the hair, face, or any part of the head in the Qur'an. God expects women to dress modestly and specifies that the chest area should be covered in public. He has deliberately allowed these guidelines to be left open within the principles of modesty in order to provide a flexible and adaptable approach to differing lifestyles. There is also indication in the Qur'an that guidelines on women's dress code should not take precedence over their righteousness and faith.