I have taken the time to discuss this question at length only because some contemporary jurists have begun issuing blanket rulings that saʿy may precede ṭawāf, and Allah alone is the One whose help is sought.

[Note]
If a woman completes her ṭawāf and then begins menstruating, she should still perform the saʿy, for ritual purity is not a prerequisite for saʿy. Al-Mughnī (5/246) states: “Most scholars hold that saʿy between Ṣafā and Marwah does not require ritual purity. Among those who said so are ʿAṭāʾ, Mālik, al-Shāfiʿī, Abū Thawr, and the Ḥanafī scholars.” He then says: “Abū Dāʾūd reported: I heard Aḥmad say, ‘If a woman finishes her ṭawāf of the House and then menstruates, she should perform saʿy between Ṣafā and Marwah and then depart.’ It is also related from ʿĀʾishah and Umm Salamah that they said, ‘If a woman has completed her ṭawāf of the House, prayed two rakʿahs, and then menstruates, let her do the saʿy between Ṣafā and Marwah.’” This was narrated by al-Athram. End quote.

[Women may leave Muzdalifah with the weak after the moon has set]
12 – Women are permitted to depart Muzdalifah together with the weak after the moon sets and to stone the Jamrat al-ʿAqabah as soon as they reach Minā, in order to spare them the crowds.
Al-Muwaffaq said in al-Mughnī (5/286): “There is no objection to sending the weak and the women ahead. ʿAbd al-Raḥmān ibn ʿAwf and ʿĀʾishah used to send the weak members of their households ahead, and the same ruling was given by ʿAṭāʾ, al-Thawrī, al-Shāfiʿī, Abū Thawr, and the Ḥanafī scholars. We know of no dissenting view. The reason is that it is easier for them and spares them the hardship of the crowd.”