The Quran and Sunnah state that a woman should cover everything. The head and hair are included in everything. Evidence is not required to prove that they are must be covered, rather evidence is required to prove the claim that they are an exception, and no such evidence exists.
The evidence of covering everything, including the head\hair\neck includes:
Quran:
What to cover?
Surah al-Ahzab, verse 59
يا أيها النبي قل لأزواجك وبناتك ونساء المؤمنين يدنين عليهن من جلابيبهن
O Prophet, tell your wives and your daughters and the women of the believers to bring down over themselves [part] of their outer garments.
— Quran 33:59
This verse states that women should bring down over themselves their Jilbab. "Themselves" means their entire selves by default, including their heads and hair. There is no exception mentioned in this verse and hair are also not mentioned as an exception in any other verse or hadith so they are not an exception.
Further the garment called Jilbab is by definition a garment that covers the entire body and especially the head.
والجلباب في لغة العرب التي خاطبنا بها رسول الله ﷺ هو ما غطى جميع الجسم لا بعضه
In the language of the Arabs with which the Messenger of Allah addressed us with the Jilbab is a garment that covers the whole body and not just part of it
— Al-Muhalla bi’l-Aathaar
جلباب ... والصحيح أنه الثوب الذي يستر جميع البدن
Jilbab ... what is correct is that it is a garment that covers the entire body
— Tafsir al-Qurtubi
Further, covering the head is how the Muslims applied this verse when it was revealed.
عن أم سلمة، قالت لما نزلت { يدنين عليهن من جلابيبهن } خرج نساء الأنصار كأن على رءوسهن الغربان من الأكسية
Narrated Umm Salamah:
When the verse "That they should cast their outer garments over their persons" was revealed, the women of Ansar came out as if they had crows over their heads by wearing outer garments.
— Abu Dawud
Surah al-Ahzab, verse 53
وإذا سألتموهن متاعا فاسألوهن من وراء حجاب
And when you ask for something, ask them from behind a partition
— Quran 33:53
This verse states that when men have a need to visit non-mahram women in their homes then there must be a curtain between the two. This means that a woman's entire body has to be covered, and hair is included. There is no mention of any exceptions in this verse rather the apparent meaning is that everything is concealed behind the curtain including hair.
When the hair are covered from visiting non-mahrams inside the house they must also be kept covered when the woman goes outside.
Surah al-Noor, verse 31
وقل للمؤمنات ... ولا يبدين زينتهن إلا ما ظهر منها
And tell the believing women ... to not expose their adornment except that which [necessarily] appears thereof
— Quran 24:31
This verse commands women to conceal their adornments with some exceptions. This literally means that they have to conceal all adornments unless they are mentioned as exceptions.
Hair are an adornment: They are combed, styled, dyed and decorated with jewellery like tiaras and chains etc.
Hair are not explicitly mentioned as an exception in this verse or in any other verse, nor in the sunnah. There is however an exception mentioned in the verse which is "that which [necessarily] appears thereof " This means:
- What is not possible to conceal i.e. outermost layer of clothing. Hair can be concealed easily with at least one layer of clothing.
- What has to be uncovered due to some necessity. Such as the eyes for seeing where one is going, or the face for verification of identity. There is nothing which necessitates uncovering the hair.
- What gets uncovered accidentally such as during movement. This does not mean that it can be left uncovered intentionally.
Hence Hair are not among the exceptions, and they are included by default in what must be concealed.
The verse goes on to say:
وليضربن بخمرهن على جيوبهن
and to wrap [a portion of] their headcovers over their chests
— Quran 24:31
This part of the verse commands to also cover the chest with their headcover. This automatically implies that a headcover must also be worn.
Obligation?
The above verses command to cover and a command is by default considered an obligation. Further the fact that it is an obligation is also proven by:
Surah al-Ahzab, verse 55
لا جناح عليهن في آبائهن ولا أبنائهن ولا إخوانهن ولا أبناء إخوانهن ولا أبناء أخواتهن ولا نسائهن ولا ما ملكت أيمانهن واتقين الله إن الله كان على كل شيء شهيدا
There is no blame upon women concerning their fathers or their sons or their brothers or their brothers' sons or their sisters' sons or their women or those their right hands possess.
— Quran 33:55
This verse is evidence that Hijab is obligatory. Because it says that there is no blame on uncovering in front of mahrams, this implies that there is blame for uncovering in front of non-mahrams. When there is blame it implies that a person is abstaining from something obligatory since there is no blame on abstaining from something that is merely recommended or permissible.
Surah al-Noor, verse 60
والقواعد من النساء اللاتي لا يرجون نكاحا فليس عليهن جناح أن يضعن ثيابهن غير متبرجات بزينة وأن يستعففن خير لهن
And women of post-menstrual age who have no desire for marriage - there is no blame upon them for putting aside their outer garments [but] not displaying adornment. But to modestly refrain [from that] is better for them.
— Quran 24:60
This verse is also evidence that Hijab is obligatory. Because it says that there is no blame on old women who remove their outer garment, which implies that there is blame on young women who remove it. When there is blame it implies that a person is abstaining from something obligatory since there is no blame on abstaining from something that is optional.
Sunnah:
Hadith of Abdullah Ibn Masud
إن المرأة عورة
A woman is 'Awrah (that which must be concealed)
— Saheeh Ibn Khuzaimah
This hadith states that a woman (in her entirety) is to remain concealed. The head is not mentioned here or in any other hadith as an exception and so it is not an exception.
Hadith of Aisha
إن المرأة إذا بلغت المحيض لم تصلح أن يرى منها إلا هذا وهذا وأشار إلى وجهه وكفيه
The Messenger of Allah (ﷺ) said: ... When a woman reaches the age of menstruation, it does not suit her that she displays her parts of body except this and this, and he pointed to his face and hands.
— Abu Dawud
This hadith proves that a woman must cover everything and some exceptions are mentioned (face and hands). The head or hair are not mentioned as exceptions and so they are not exceptions and must be covered.
Hadith of Fatimah
أن النبي صلى الله عليه وسلم أتى فاطمة بعبد قد وهبه لها قال وعلى فاطمة رضي الله عنها ثوب إذا قنعت به رأسها لم يبلغ رجليها وإذا غطت به رجليها لم يبلغ رأسها فلما رأى النبي صلى الله عليه وسلم ما تلقى قال إنه ليس عليك بأس إنما هو أبوك وغلامك
The Prophet (ﷺ) brought Fatimah a slave which he donated to her. Fatimah wore a garment which, when she covered her head, did not reach her feet, and when she covered her feet by it, that garment did not reach her head. When the Prophet (ﷺ) saw her struggle, he said: There is no harm to you: Here is only your father and slave.
— Abu Dawud
The above hadith is about the relaxation of hijab in front of mahrams like a woman's father and slave. The Prophet ﷺ said "There is no harm to you: Here is only your father and slave" ... this implies that there would be harm if someone other than her mahrams were present, which means that hijab from non-mahrams is obligatory. Further this hadith explicitly proves that covering the head is part of hijab.
Ijma:
The jurists all agree that the head and hair must be covered. Consensus is evidence by itself in Islamic law.
واتفقوا على أن شعر الحرة وجسمها حاشا وجهها ويدها عورة
They agreed that the hair of a free woman and her body except her face and hands is to be covered
— Maraatib al-Ijmaa‘