New leadership role and handling disgruntled employees.

Hi Brains trust, I was recently hired into a new job(yay for me) the role is a team lead position and will have a team of about 5 most of the time.

Here is my concern, as mentioned I am an outside hire and there was 2 employees on my team that went for the role but were unsuccessful.

I spoke with my new boss the other day and he has mentioned one of the guys is unhappy to have been overlooked and is already starting to make his opinion known.

My question is how do I get ahead of this situation? Do I speak to him directly first up or wait it out ?