At first it should be mentioned:
Our sources in Islamic precepts are: 1- Quran, 2- Hadithes and 3- Reasoning.
If we believe in Quran, - that surely we believe - Allah said:
1- قُلْ أَطیعُوا اللَّهَ وَ الرَّسُولَ فَإِنْ تَوَلَّوْا فَإِنَّ اللَّهَ لا یُحِبُّ الْكافِرین
Say," Obey Allah and the Prophet. " But if they turn away, indeed Allah does not like the faithless.
Quran: 3: 32.
2- وَ أَطیعُوا اللَّهَ وَ الرَّسُولَ لَعَلَّكُمْ تُرْحَمُونَ
and obey Allah and the Prophet so that you may be granted] His [mercy.
Quran 3: 132.
3- قُلْ أَطیعُوا اللَّهَ وَ أَطیعُوا الرَّسُولَ فَإِنْ تَوَلَّوْا فَإِنَّما عَلَیْهِ ما حُمِّلَ وَ عَلَیْكُمْ ما حُمِّلْتُمْ وَ إِنْ تُطیعُوهُ تَهْتَدُوا
Say," Obey Allah, and obey the Prophet." But if you turn your backs, (you should know that) he is only responsible for his burden and you are responsible for your burden, and if you obey him, you shall be guided.
and you can refer to: Quran 4:59, 5:92, 8:1, 8:20, 8:46, 20:90, 24:54, 24:56, 47:33, 58:13, 64:13.
So if we have any tradition (Hadith) without Quranic verses, we should follow the tradition of our prophet.
Secondly: We have a lot of Hadithes about covering hair.
Thirdly: About the obligatory of covering hair in Quran, we have some verses, two of them are:
1- وَ قُلْ لِلْمُؤْمِناتِ یَغْضُضْنَ مِنْ أَبْصارِهِنَّ وَ یَحْفَظْنَ فُرُوجَهُنَّ وَ لا یُبْدینَ زینَتَهُنَّ إِلاَّ ما ظَهَرَ مِنْها وَ لْیَضْرِبْنَ بِخُمُرِهِنَّ عَلى جُیُوبِهِنَّ وَ لا یُبْدینَ زینَتَهُن
And tell the faithful women to cast down their looks and to guard their private parts, and not to display their charms, except for what is outward, and let them draw their scarfs over their bosoms, and not display their charms .
Quran 24: 31
2- یا أَیُّهَا
النَّبِیُّ قُلْ لِأَزْواجِكَ وَ بَناتِكَ وَ نِساءِ الْمُؤْمِنینَ
یُدْنینَ عَلَیْهِنَّ مِنْ جَلاَبِیبِهِنَّ ذلِكَ أَدْنى أَنْ یُعْرَفْنَ
فَلا یُؤْذَیْنَ وَ كانَ اللَّهُ غَفُوراً رَحیماً
O Prophet! Tell your wives and your daughters and the women of the faithful to draw closely over themselves their chadors] when going out [. That makes it likely for them to be recognized and not be troubled, and Allah is all-forgiving, all-merciful
Quran 33: 59 .
The way of proof:
وَ لْیَضْرِبْنَ بِخُمُرِهِنَّ عَلى جُیُوبِهِن
And let them draw their scarfs (Khimar) over their bosoms. Quran 24:31
At the age of prophet, women cover their hair by scarf (Khimar) and Allah says: in addition of that, let them draw their scarfs (Khimar) over their bosoms.
So covering of hair was obliviously obligatory, and women covered their hair naturally but in addition Allah obligate covering of bosom and charms.
In other word, they do three steps of below picture, and Allah commend them to do others steps.
1- we should follow Hadith too, and we have a lot of hadithes about covering hair.
2- Allah said: Women in addition of covering their hair with khimr, they cover their bosoms too.