Long-lived particles are a compelling direction in which to search for physics beyond the Standard Model, and implementing dedicated long-lived particle (LLP) triggers provides an excellent avenue to expand experimental coverage into this challenging parameter space. We present a novel Compact Muon Solenoid (CMS) Level-1 LLP trigger that exploits the Run 3 upgrade of the Hadron Calorimeter (HCAL), which introduced a precision timing ASIC, programmable front-end electronics, and depth segmentation to the CMS HCAL barrel. The hardware- and firmware-based trigger algorithm identifies delayed jets, resulting from the decay of massive LLPs, and displaced jets, resulting from LLPs that decay inside the HCAL. This approach significantly increases sensitivity to LLP signatures with soft hadronic final states, including exotic decays of the Higgs boson. Simulated LLP events demonstrate the sensitivity to displaced jets, with high trigger efficiency for LLPs decaying in the calorimeter volume. Recent HCAL timing scans provide a valuable look at artificially delayed jets in collision data and are crucial to understanding the trigger performance. The data collected with the new triggers implemented for Run 3 provides a first look at the capabilities to capture softer events and expand the phase space accessible in ongoing LLP searches.