It may have taken tens of thousands of years after leaving Africa for modern humans to establish permanent settlements beyond Southwest Asia, but modern humans left Africa at least about 185,000 years ago. And, even then, the range expansion appears to have been driven by technological advances.
To date, the earliest modern human fossils found outside of Africa are dated to around 90,000 to 120,000 years ago at the Levantine sites of Skhul and Qafzeh. A maxilla and associated dentition recently discovered at Misliya Cave, Israel, was dated to 177,000 to 194,000 years ago, suggesting that members of the Homo sapiens clade left Africa earlier than previously thought. This finding changes our view on modern human dispersal and is consistent with recent genetic studies, which have posited the possibility of an earlier dispersal of Homo sapiens around 220,000 years ago. The Misliya maxilla is associated with full-fledged Levallois technology in the Levant, suggesting that the emergence of this technology is linked to the appearance of Homo sapiens in the region, as has been documented in Africa.
Israel Hershkovitz, "The earliest modern humans outside Africa" 359 (6374) Science 456-459 (January 26, 2018) DOI: 10.1126/science.aap8369