I think that as long as you understand the client’s goals and any limitations you will be fine! Chances are, you will train differently than the previous trainer and will have your own unique style. That alone may provide a challenge, or even progression, to your client’s current program (of course understanding the client’s limitations). Before I would worry about progressing the client, I would just take some time to make a connection and get to know their personality and movement pattern.
You may provide some new exercises and challenges that will help the client feel motivated and good about exercising. Keep setting those SMART goals and make the sessions fun–have the client try some new exercises they have not done before. As you work with the client you will become more comfortable in progressing him or her.