Is there more than one triggering mechanism for the Jews' flight to the mountains?

Yes, there is more than one triggering mechanism. The context identifies an additional triggering mechanism in Revelation that causes Jews to flee from Jerusalem down to the mountains to the southwest, where a remnant from every category of life is protected by God for three and a half years.

Les explains that Revelation reveals another triggering mechanism that will cause these Jews to go down to the mountains, which are predominantly southwest of Jerusalem. He calls this the Escaping Remnant—not the whole nation, but a remnant of Jews from every category of life. They will flee the area of Jerusalem down to the mountains, possibly Petra, though he does not know for certain. Their God will watch over and protect them for the whole three and a half years. The context also notes that the tribulation period includes both Satan's wrath and God's wrath, making it severe.

What will trigger the Jews to flee to the mountains southwest of Jerusalem?

The tribulation described in Matthew 24:21, when Satan's wrath and God's wrath are poured out on the human race, will trigger the Jewish remnant to flee from Jerusalem to the mountains southwest, where God will protect them for three and a half years.