In the Run-3 operation of the LHC from 2021, the center-of-mass energy is planned to be increased to 14 TeV, and the instantaneous luminosity is maintained at about $2.0 \times 10^{34}$ cm$^{-2}$ s$^{-1}$. In anticipation of a further increase of the luminosity in the future, an upgrade of the ATLAS trigger system is ongoing during the shutdown between Run 2 and Run 3. The level-1 endcap muon trigger system identifies high-$p_{\textrm{T}}$ muons based on a coincidence of the signals from fast muon trigger detectors, Thin Gap Chamber (TGC), and the tile calorimeter (TileCal). From Run 3, in addition to them, new muon detectors, New Small Wheel (NSW) and RPC-BIS78, will be installed to provide better means for fake rejection and momentum resolution of the muon trigger. A combination of signals from TGC, TileCal, NSW, and RPC-BIS78 and wider $\eta$ coverage of NSW can improve the performance of the level-1 endcap muon trigger significantly. A trigger processor, Sector Logic (SL) board, will be fully upgraded, which collects the signals from the four different detectors, performs a fast reconstruction of the muon candidates and passes the results to an interface board between muon trigger systems and the central trigger processor (CTP), called MUCTPI. An increased number of logic gates and the new high-speed optical interface make it possible to send much more information to MUCTPI, for example, $p_{\textrm{T}}$ measured in 15 steps from 4 GeV to 20 GeV, muon charge information, and some trigger condition flags. The expected efficiency is about 97% for muons with $p_{\textrm{T}}>20$ GeV compared to the Run 2 performance, while the trigger rate is reduced down to 8--13 kHz depending on the configuration of NSW. The installation of the new SL boards has been completed, and the on-site commissioning is in progress.