The Hadron Calorimeter (HCAL) in the Compact Muon Solenoid (CMS) experiment at the Large Hadron Collider (LHC) was recently upgraded for Run 3 to introduce depth segmentation and online timing measurements, expanding the physics capabilities. In particular, the augmentation of the calorimeter information at the hardware trigger level enables quick identification and recording of long-lived particle decays using lower thresholds on energy-based event quantities. The depth segmentation and online timing capabilities are utilized in novel HCAL-based hardware-level triggers to identify displaced and delayed long-lived particles (LLPs), either decaying inside the calorimeter volume or arriving at a delayed time. This increases the sensitivity to LLP decays occurring up to almost 6 m from the collision point. This two-pronged calorimeter trigger approach leverages the new capabilities of the CMS HCAL to expand the phase space accessible in ongoing LLP searches. These triggers were deployed for Run 3 of the LHC, beginning data-taking in 2022, and we review the trigger implementation and calibration.