How do I break into an entry-level strategy role?
Business intuition and creative thinking are only part of the expectations for junior strategy candidates.
Analytical rigor is typically table stakes for these types of roles, especially at entry levels where you may be competing with Business Analyst-types. Junior strategists often run the financial and operational modeling and the detailed quant behind the opportunities that are identified, sized, and recommended. Additionally, strategy positions typically involve case interviews where your interviewers will want to test your comfort and speed with quantitative analysis.
If you have experience you can allude to, make sure you speak to your ownership of those models and analyses. If not, then I would find a way to start doing that type of work.
General advice is to tell as many people as possible that you want to make that career change. I see time and again that if you are successful in your current role, make it clear what you want next then opportunities will come to you.
More specifically, in larger organisations the two main ways I have seen people shift into new roles is:
1. Secondments. Look for internal mobility opportunities to try the role out of a while. Often hiring managers are more open to taking on an entry level person when the role is time bound like covering someone's leave or a specific project. By demonstrating your success in a short term role you have more chance of being offered a full time position.
2. Side projects. Seek out projects that are not core to your day job but that allow you to demonstrate your potential in other areas such as strategy. Many organizations have hack days or 20% time that allow you opportunities to show your other skills and interests.