As an agile coach you would like to help your team from a place where they need to build and strengthen their Agile Capabilities to a place where they have strong agile capabilities and have achieved more Agility.
When we as coaches enter new teams it is because teams want to strengthen their agile capabilities, they might want to have more consistent processes and more consistent outcome.
Most teams also want to follow their own path to agility because there are many things in their current processes that work well, and there would be no need to do changes on things that work well. Listening to the team and understanding their current situation will give you as a coach a good start.
From an agile coaching perspective it could be good to think about what journey you want to bring the team on so that they reach to where they want to be when it comes to skills, experience, capacity, knowing their options and increase their decision making frequency.
Initially a lot of focus an effort needs to be put into understanding the current situation, what did the team already try and where do they want to go?
Understanding their current situation is also about alignment within the team on where they actually are. I find it useful to give the teams some initial training at this point such as an introduction to Agile and also learning some flow mechanics and principles so that they have a better understanding of what is possible and where they could be headed. This also helps in understanding what could be improved.
After learning where the team is, what they want to achieve and where they see challenges and waste in their system it is time to define some experiments, perhaps even an improvement backlog and start working on those improvement, trying, learning and iterating on these until the consistency in process is well defined and achieved.
After the team has well defined and consistent processes the outcome starts to become more consistent and the team becomes more predictable.
As a coach you will continuously assess the teams maturity level. It could be a an idea to look into how to establish a baseline and how to measure progress. How will you know that the team is improving overall on their journey?