Talk to people. Talk to the gym members. Get to know the gym members as friends.
Tell everyone you know/meet that you’re a personal trainer and where you train.
Also, don’t worry about sounding like a salesperson. Personal training is sales. Confidence is the key. Just talk to people about yourself, why you train, the benefits of, and how you want to help them.
The best way to build relationships directly off the gym floor is to correct people, mid-exercise, on how to better perform a movement. The lat pull-down and leg press seem to be the most commonly abused exercises. Speak to how they can make the exercise more efficient while reducing the possibility for injury. Where the conversation goes from there is all up to you. You can ask them outright if they have ever worked for a trainer, how was it or why not, speak to how much more you can show/teach them, get to know their fitness goals, etc.