However Saudi Arabia said the disaster-prone Jamarat Bridge in Mena, where the stampede occurred and the scene of several similar tragedies in the past, will be demolished next week ahead of a major renovation.
A spokesman for the interior ministry said tighter controls on the massive crowds could have led to further deaths.
"The total death toll has reached 363 so far," interior ministry spokesman Mansur al-Turki said, as stricken families continued to hunt for their loved ones.
Rows of photographs of unidentified bodies have been posted on a wall and a television screen at the Muayasem morgue.
Mr Turki said 203 bodies have been identified, 118 of them male and 85 females.
He blamed the masses of pilgrims crowding to perform the stoning ritual at the same time around mid-day, as they aimed to leave the area before sunset.
"The problem happens when some 2.1 million pilgrims decide to leave Mina (at the same time)," Turki told reporters, saying that this figure represents 70 percent of the total official and irregular pilgrims.
Mr Turki said the dead pilgrims include 10 Egyptians, 44 Indians, 37 Pakistanis, 11 Begalis, five Algerians, 18 Saudis, seven Yemenis, six Turks, six Sudanese, six Maldivians, four Moroccans, five Afghanis, three Iraqis, two Chinese, two Syrians, one Chadian, one Belgian, one German, one Nigerian, one Jordanian, and one Ethiopian.
Officials earlier said the death toll included around 100 Egyptians and 70 Asians.
While authorities said pilgrims must be better behaved and know the rules and practices of the hajj.
Many pilgrims insist on following Prophet Mohammad's example of stoning after midday prayers, rather than stagger the ritual throughout the day, as some clerics have recommended.
Thursday's stampede is the worst during hajj since 1,426 pilgrims were killed in 1990.
In 2004, about 250 pilgrims were crushed to death at Jamarat Bridge, and a decade earlier, 270 were killed in a similar stampede.
Almost 60,000 security, health, emergency and other personnel were involved in organising this year's hajj, trying to prevent the deadly incidents that have marred it in recent years from being repeated.
The hajj, which follows a journey by Prophet Mohammed over 1,400 years ago, is one of the five pillars of Islam and a once-in-a-life time duty for those able to complete it.