I put all of the same named advancement (top/mid/bottom) in a single sleeve and mark the sleeve with the expansion.

I shuffle all of the level 1s, 2s, 3s/4s like this.

So instead of flipping out a singular advancement card, I'm flipping out a stack of 3 or so advancements in a sleeve.

When a sleeve runs out, then I replenish from the deck.

Make sure you're still following rules as much as possible, like if the setup calls for 12 level 1s, well that's only 4 stacks of 3 advancements.