Scrum Master job by default starts when I felt like a catalyst my organization needed. I have constantly worn the hat of Project Manager and Scrum Master which are two different roles. With any new client, I like to understand the environment and the need that triggered to hire me. That helps me to create my mission statement.
My usual approach with a new client is to ensure Scrum Framework becomes part of their work culture. If they already have the framework integrated then I ensure the 3 categories of this framework - Scrum Team, Scrum Events, and Scrum Artifacts have been established and are in use properly.
So far every client I have worked for, even though they want to adopt Agile, but very few could adopt it perfectly. The reason is change management and how the change is being handled by the entire company. I have worked with Product Owners to ensure the company is aware of critical change management requirements.