Yes, the Prophet (peace be upon him) explicitly granted a concession for weak and vulnerable pilgrims to leave Muzdalifah after midnight, before the Fajr prayer. He permitted his wife Sawda bint Zam'ah and other members of his family who were weak to leave early and proceed to Mina ahead of the main body of pilgrims. This was done to protect them from the crush of crowds during the mass departure at dawn.
This concession applies to: elderly pilgrims, those with illness or physical weakness, women (especially those who may face difficulty in crowds), young children, and the companions or caregivers of any of these groups. The caregiver's concession is important — if an elderly person needs assistance, their helper may also leave early.
Those who depart early should proceed directly to Mina and perform the stoning of Jamarat al-Aqabah (the largest pillar) before the main crowd arrives. This is actually a significant practical advantage, as the stoning area becomes extremely congested later in the day. The minimum time threshold for the concession is after midnight — leaving before midnight without a genuine emergency would mean missing the wajib of the Muzdalifah stay, which requires compensation (fidyah).