How can I get another member of The Companions to join me?

I completed the companions mission, and asked for one of them to follow me, but they died in a cave. When I went back to the companions to get a new one they said I already had someone.

How do I get another companion?

Are you sure that your companion follower is dead? Was it something like they fell off a ledge and you think they died? One of my followers who was dismissed fell into the pool outside dragonreach when going back home and never got out. I assumed she died until I fell in the pool a few days later and noticed her just standing underwater.

Furthermore, by my experience, companions guild followers cannot die as they are quest characters; The worst that can happen is they are stunned and crouch down for a few seconds to regain their strength (I know this for a fact because I tried to kill Aela when she was following me and was unable to; I could easily kill Lydia or Jenassa).

Seems like your follower is stuck somewhere and you need to find him/her.

If the person really did die, try doing a few quests and then coming back. Usually any glitches I have get fixed when I do some other quests.