I'm trying to work that into the world arc I'm building.
T1 is introduction...
T2 is intended to be the repercussions of that thing of great import...
By T3, the once they find a dungeon, grabbing the shiny should be almost free. So two big changes happen.
First, the cold war that has been brewing goes hot and rapidly becomes a world war. The PCs will hopefully made friends with or give a naughty word about at least some NPCs or organizations, and those organizations will be involved or threatened by said war.
Second, the trail of great import + shinys point directly at an outside context problem...
T4 is about the outside context problem coming to a head. Success is not an option, mitigation is....
That is very much like how I approach campaign construction. I think of it as the Babylon 5 method, as I borrow heavily from the way that JMS structured that 5 year story (with twists):
There is a ALPHA STORY, 2 or 3 BETA STORIES, and 5 to 10 MINOR STORIES.
The ALPHA STORY will only be hinted at in the first tier of play, but will be explored in tiers 2 and 3 amongst the MINOR STORIES, but will climax at the end of the campaign. This is
THE Story of the campaign, although PCs will spend a long time coming to understand that element. Referencing back to Babylon 5, this would be the War of the Shadows (which didn't end the show, but it was the major story of the show).
The BETA STORIES will either be substories in the ALPHA story, or they'll be independent. Either way, they span long periods of time and impact a lot of aspects of the game. They often provide framework for Minor Stories. Using Babylon 5, the Centauri/Narn war and the Earth Civil War are examples of Beta Stories. These BETA STORIES will resolve in a minimum of 8 PC levels of story development.
The MINOR STORIES are generally adventures. If you want to think of it in terms of Babylon 5, they'd represent anything from one episode to a series of connected episodes, but no more than 4 or so. The Babylon 4 Story Arcs, the Exploration of the planet, the Psi Corp episodes - each would be minor stories. The game starts with MINOR STORIES, and there are always a few options for MINOR STORIES that will give the PCs information they need for the BETA and ALPHA STORIES.
These unfold through the tiers of play in ways similar to what you discuss.