Already some good information here. Talking to people is the best way to get them to know you. Also offering a few good tips here and there to members will also go long way. Other options would be to join different networking groups to get some leads from there as well. I’m sure by now you have figure out how to get new clients :-)!
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.
Do complimentary sessions. As a new trainer I did lots of comp sessions over the years and gained many clients from this interaction. Most clubs, as part of new member package, offer 1 to 3 complimentary sessions. Call the new members and schedule these sessions for them. Familarize them with the components of fitness, while you show then how to use the equipment. This starts building rapport. Call them by their name, and be genuinely concerned about their goals and interests. Be a partner in their quest. If this is not an option at your gym discuss the possibility with the club owner.
If floor time is an option, do that as well. This gives you a “raison etre” and opens you up to member connection.
Try setting up a desk during high traffic time and offering free bodyfat testing, or any other fitness component testing, such as sit and reach. All easy, all creating recognition, passion and connection. Give something of value.
Have a contest and the winner gets ___ free sessions with you.
Be creative and make it fun.
All the Best