In the specific case of this article, they're interested in particles that look a lot like a muon. The signature of a muon in a particle detector is that it leaves a charged track in the inner tracker, passes through the substantial mass of the electromagnetic calorimeter without saying much, and then goes pow in the muon detector (usually alternating plates of steel and detector).
A long-lived particle could be quite similar; a charged track, a lot of nothing, and a decay in the muon tracker.
Any particle that might roughly replicate the muon's signature might have hidden from existing searches for new physics. The proposal in the article is to cover the muon detector with higher-resolution trackers. If the "muon"'s decay doesn't conform to the standard plan, higher resolution would make it apparent.
I'm not enough of an expert to know how sensitive this method is in comparison to other ways of hunting long-lived particles, but I have taken a class or two from one of the authors. If Lubatti's pitching the idea, it's worth thinking about.