The Phase-2 upgrade of the CMS Level-1 (L1) trigger will be carried out to achieve the highest possible acceptance for physics processes in the large pile-up and instantaneous luminosity conditions of the High-Luminosity Large Hadron Collider (HL-LHC). In the CMS endcap region, muon reconstruction in the L1 trigger is complicated by the non-uniform magnetic field, high pile-up and punch-through interactions. However, the new muon detectors and the upgraded trigger processing capabilities proposed for the Phase-2 upgrade will allow the implementation of novel techniques to address successfully these challenges. For instance, track-finding and reconstruction of the standalone and displaced muons are performed by a neural network-based algorithm. In addition, a proposed Global Muon Trigger system will have access to tracker tracks, muon trigger tracks and individual muon detector hits. These objects can then be combined to improve the muon momentum resolution, reduce the muon trigger rates, and form multi-object triggers targeting specific processes such as charged lepton flavour violating $\tau\rightarrow3\mu$ decays. We present here preliminary studies addressing all these new capabilities.