Iran using their proxies to spread terror around the globe pre-dates Gulf War II. The mismanagement of the immediate aftermath of the invasion along with the decision to just pull up stakes and leave without first vastly reducing Iran's ability to influence/destabilize the new Iraqi government didn't help.

Of course, but prior to Gulf War II the other nations in the region didn't fear Iran as much as they do now. With Iraq as another regional hegemon hostile to Iran, Iran was more constrained in what it could do - particularly against the Sunni states, with a massive Sunni-controlled military on their borders. So while they could foment disruption against Israel (whom few in the region loved), direct support of proxies attacking other Sunni nations was far more dangerous. Iran wants all of the secular Sunni governments in the region to be replaced by Islamic ones (whether Shia or Sunni), but faced a greater natsec risk in taking measures to make that happen while Iraq was still an enemy.

The fall of Iraq opened up the sandbox for them, as it were. The secular governments in the region are threatened by Islamist activism (one reason El-Sisi despises the Muslim Brotherhood), so Iranian support for Islamist movements represents a real danger for them. Which is why they've all been cozying up to Israel recently, since they pose the only real military danger to Iran comparable to what was lost when Iraq switched sides.