The Level-1 endcap muon trigger in the ATLAS experiment is a hardware trigger system for the endcap region. The system identifies bunch crossings with muon candidates of high transverse momentum by combining information from fast-response muon detectors. The Large Hadron Collider at CERN is scheduled to start the physics data-taking from 2021 as Run 3. An upgrade program of the Level-1 endcap muon trigger for Run 3 is required to reduce backgrounds in the muon trigger candidates. New muon detectors called New Small Wheel, and RPC-BIS78 will be installed to improve detector coverages and transverse momentum resolution. Thus, a new trigger processor board called Sector Logic has been developed to handle information from the new detectors. This paper describes the upgrade program of the Level-1 endcap muon trigger, particularly emphasizing on the new algorithm on Sector Logic board as well as the expected trigger performance.