The standard sequence in both হজ্জ and উমরাহ is তাওয়াফ first, then সাঈ. For উমরাহ, all four schools agree that সাঈ must follow তাওয়াফ — this sequence is an integral part of the ritual. Performing সাঈ before তাওয়াফ in উমরাহ would mean you need to redo the সাঈ after completing তাওয়াফ.
For হজ্জ, there is more scholarly discussion. The Hanbali school allows some flexibility: if a হাজী has already performed তাওয়াফ al-Qudum (the arrival তাওয়াফ), they may perform সাঈ after it. They may also, according to some Hanbali scholars, perform the সাঈ of হজ্জ on the Day of আরাফাত before performing তাওয়াফ al-Ifadah. This is based on the hadith where নবী করীম (সাল্লাল্লাহু আলাইহি ওয়া সাল্লাম) was asked on the Day of Sacrifice about performing rituals out of order and said: 'Do it, and there is no harm.'
The Hanafi, Maliki, and Shafi'i schools maintain that সাঈ must always follow a preceding তাওয়াফ, whether it is তাওয়াফ al-Qudum or তাওয়াফ al-Ifadah. If someone performs সাঈ without a preceding তাওয়াফ, it is not counted and must be repeated. For practical purposes, following the standard sequence (তাওয়াফ then সাঈ) is the safest approach accepted by all schools. If you find yourself in an unusual situation, consult a knowledgeable scholar present at the Haram.