As for the details regarding staying overnight in Mina while enjoying Hajj rituals, the pilgrim is required to stones the three Jamarat in the way beginning with the first Jamrah al-Khaif which is the nearest to Masjid.

After stoning al-Khaif, you are required to move onward, stands facing the Qiblah (The Kaba) for a long and making du'a along with raising your hands to the sky. The move to second jamarah and repeat every step you took while and after stoning the first one, except moving to the left while facing the Qiblah.

Finally, move to the third one, which is called Jamrat al-Aqabah, and stone on the third Jamara in same way you stoned to the first Jamara. Standing in a way that Ka'bah and Qiblah (Situated in Mecca) to your left and Mina to your right and does not stand there (making du'a) afterwards. And you are to repeat this stoning on second and third day in a same way as a manasik hajj from the Prophet of Islam Hazrat Muhammad SAAW.

However remaining for the third day's stoning is better and not compulsory if it is done n second day of third day, as it is the Sunnah. Allah says,

“Celebrate the praises of Allah during the Appointed Days, but if anyone hastens to leave in two days, there is no blame on him, and if anyone stays on, these is no blame on him if his aim is to do right,”

Therefore, remaining for the third day's stoning is better as it is the Sunnah. So Sunnah is to follow all he previous action in same order as Rasool Allah SAAW did, however, there is no harm to change the order of action of these rituals of hajj (Pilgrimage) as Prophet SAAW said,
 

“There is no harm, there is no harm.”