When the match was about to end, I Alt+F4 and waited for the honor screen.
I went play another one. Picked my champion. Waited for the time.
The Loading Screen didn't show.
I restarted the Client. But nothing happened.
I went over to investage and the "LeagueofLegends.exe" process was running and using CPU.
When I killed the process, through the task manager, the Reconnect button showed up and I was able to join the match.