When should you start working at the organizational level?

From day one. Anything less devalues the Scrum Master as we see all too often in the job market.

The Scrum Master is not an entry-level position. Facilitation, coaching, mentoring, teaching, being a change agent, causing the removal of impediments, and knowing more than Scrum are some of the skills acquired over time. Yes, you can be a beginner Scrum Master as we all have to start somewhere. Yet that doesn’t make a person taking on Scrum Master accountabilities beginner-level.

Certainly a Scrum Master with a team just starting out will spend more time with the team. That doesn’t mean ignoring the Product Owner and organization. 

A Scrum Master, or any other lean-agile coach, should be ready to work effectively on all levels from the start.

To start with, a Scrum Master should always be an experienced individual. They may not necessarily be experienced at being a full-time coach, but coaching isn't an entry-level position. When it comes to the skills needed, I tend to use the Agile Coach Competency Framework as a foundation. As a base, a Scrum Master should have deep experience in technical skills (likely as a Developer on agile, if not Scrum, teams) or business (likely as a Product Owner or otherwise working in product management) or organizational leadership and transformations (with a background in industrial/organizational psychology, organizational behavior, or as an organizational leader or in HR). Having recognized deep experience and expertise in at least one of these areas is key, and having exposure to the other areas is greatly beneficial. Since they are experienced in at least one of these domains, they should have had the opportunity to teach and mentor less experienced individuals and have been involved in various ways to develop their skills. They should have also demonstrated some experience in facilitating and coaching. If at least some of this practice - the mastery of a relevant domain, teaching and/or mentoring, and coaching and/or facilitating - has been done in an environment that embraces the Lean and Agile values and principles, then the person is ready to serve as a coach. The experience enables the Scrum Master to speak with authority on specific topics while recognizing the limits of their knowledge in others.

Now, I wouldn't expect someone new to the Scrum Master role to be an expert in coaching and facilitation. Perhaps they have had some formal training in these topics, but this is something that comes with experience. However, I would expect at least familiarity with these topics and the desire to learn. Being inexperienced with coaching or facilitation, however, wouldn't stop the Scrum Master from earning the trust of those around them or from being able to work at any level of the organization.

how can we tell if a Scrum Master is ready to work effectively on all three levels?

Some examples:

• Developer level: is the team becoming increasingly more self-managing? Does the Scrum Master coach the team toward solving their own problems, or do they always step in? When there is an issue do the Developers jump on it or wait for the Scrum Master? Does the Daily start with our without the presence of the Scrum Master?
• Product Owner level: Is the Scrum Master passive in Sprint Planning and product Backlog refinement? Are they actively helping clarify goals, value, and teaching splitting and ordering techniques?
• Org level: Are they engaging with leaders to help with systemic impediments? Are they educating stakeholders about empiricism, business agility, and the value of Scrum?

The question is: how can we tell if a Scrum Master is ready to work effectively on all three levels?

Think of it the other way around. How ready are each of these levels to work effectively with a Scrum Master? Is there a sense of urgency for change? That's the primary indicator of readiness. That's how you tell.

Bear in mind that there may be a need for engagement across the whole value chain.