An often overlooked tactic to build trust within a team is the art of advocacy. No, not advocating for a cause, instead advocating for people. Specifically, people who are not present when you are promoting for them.
We all know that humans are social creatures. As such, they tend to share various tidbits of information. As a leader, use this to your advantage. If you are trying to build a team, make it on the accomplishments and personalities of the team.
I can think back to when I took over a new facility. I knew that my employees would probably have some relationship with different work units outside of my scope of control. As such, I knew the quickest way for me to get buy-in with a new group of employees was to brag about them to other workgroups simply. I hedged a bet that if I continued to speak positively about the workgroup outside of their immediate social circle that eventually, they would hear of how their leader spoke about them in public.
I took advantage of learning specific positive traits of as many of my people as I could. Every chance I got, I brought them up to people. Completely unsolicited. Eventually, word got back to them that their leader believed in them. With the comfort of knowing they had the support of the formal leader, they became much more innovative and more committed to the work we were trying to accomplish.
As a leader, advocate for your people. Show others you value them. Believe in them so that they can believe in themselves.