Just got to this point today, here's what you do. Go to a dungeon and complete it or complete a mission. Before you leave with your partner he'll ask if you think it would be a good idea to have different leaders. Say Yes (don't know what happens if you say no) and the next day your partner won't meet you outside your house like he normally does. Go to his friend area and you'll find him. Put him on standby then you can go to the cave and evolve. You can also change leaders by going to the pokemon you want to be the leader and selecting "Make Leader".