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Motivated by the expectation that new physics may manifest itself in the form of very heavy new particles, most of the operation time of the LHC is devoted to $pp$ collisions at the highest achievable energies and collision rates. The large collision rates imply tight trigger requirements that include high thresholds on the final-state particles' transverse momenta $p_{T}$ and an intrinsic background in the form of particle pileup produced by different collisions occurring during the same bunch crossing. This strategy is potentially sub-optimal for several well-motivated new physics models where new particles are not particularly heavy and can escape the online selection criteria of the multi-purpose LHC experiments due to their light mass and small coupling. A solution may be offered by complementary datasets that are routinely collected by the LHC experiments. These include heavy ion collisions, low-pileup runs for precision physics, and the so-called 'parking' and 'scouting' datasets. While some of them are motivated by other physics goals, they all have the usage of mild $p_{T}$ thresholds at the trigger-level in common. In this study, we assess the relative merits of these datasets for a representative model whose particular clean signature features long-lived resonances yielding displaced dimuon vertices. We compare the reach across those datasets for a simple analysis, simulating LHC data in Run 2 and Run 3 conditions with the Delphes simulation. We show that the scouting and parking datasets, which afford low-$p_{T}$ trigger thresholds by only using partial detector information and delaying the event reconstruction, respectively, have a reach comparable to the standard $pp$ dataset with conventional thresholds. We also show that heavy ion and low-pileup datasets are far less competitive for this signature.