What are the best practices for a new Scrum Master in an already Agile Team?

I don't understand what it means to be "mature in their way of working".

I'd point to the very first sentence of the Manifesto for Agile Software Development:

We are uncovering better ways of developing software by doing it and helping others do it.

A team may be highly experienced in Scrum and proficient at delivering valuable work quickly, receiving feedback, and iterating on that feedback, that doesn't mean that they can't go beyond the methods and experiment with new techniques to push beyond their current capabilities.

At the end of the day, though, it's about what the team is capable of. Not all teams will be innovators or early adopters of techniques. I'd bring it to the team to see what they are comfortable with and perhaps work on their comfort of running experiments to seek continuous improvement.

As a new Scrum Master starting with an Agile Team who are pretty much mature in their way of working, what are the best practices and strategies?

How do you know they are "pretty much mature in their way of working"? What are you using to evidence this?

Regradless of the maturity of the team, as I can't judge it from your original post.

I would sit back and observe how the people communicate, who is responsible for what and many more. After having the more clearer picture you can define some measurements. At the beginning I try to have talks with developers to introduce myself and to build up the relationship and directly get feedback what is going on right now in the team.

It might be that I am a person who is also wanting to optimize and directly fix some issues, but always be patient, the environment and the people choose the speed of change, not you.

Bets regards

Alex

I just had a look into the blog of scrum.org and found this gem. 

Please have a look into this. It describes the first steps of a scrum master in the team, my previous post was nearly the same direction.

https://www.scrum.org/resources/blog/first-steps-scrum-master

As a new scrum master follow Shu-Ha-Ri

First follow scrum guide religiously.

With mature team as well, follow scrum guide, be yourself, practice integrity, apologize for your mistakes and your team will love it.

Work for better agile team as most mature teams also need betterment in terms of Agile.