Think about building evidence in three areas: impact, ownership, and client exposure. MBB recruiters will care less about how many projects you touched and more about what you actually did on them and whether your role became progressively more demanding.
Try to get onto projects where you can solve ambiguous problems, work with senior stakeholders, and own a piece of the analysis rather than only executing instructions. A promotion is useful because it signals performance, but the underlying stories matter more than the title by itself.
Networking can start well before you apply, but it should be light and genuine. You do not need to spend two years asking people for referrals. Build relationships, learn which offices and practices fit your background, and understand what experienced-hire recruiting looks like.
If you do those things consistently, you will have a much stronger profile when the timing is right. The goal is to make your current role produce evidence that transfers cleanly into consulting. A useful reference is what consulting firms look for in experienced hires if you want to go deeper on that part.