I put the word out in the community. I tell everyone I know that I am looking for a new trainer.
And I search for a trainer as if I were looking to hire one on sites like ACE, etc. Then contact those I find who I don’t already know personally (and a few that I haven’t seen in a while if applicable) to let them know that I am looking. I usually reach out casually with a simple “I am looking for a new trainer for my studio. Are you interested in talking about it? Do you know anyone that might be intereseted?”