the easiest way to transit to MBB after 1-3 years is joining another strategy consulting firm (ideally an internationally recognized brand such as Roland Berger, Oliver Wyman, ATK, etc.). This would also have the additional benefit to let you start in the environment you would like to move later, allowing you to (1) clarify if consulting may be the right path for you in the medium/long term and (2) potentially stay with that company if you find a great environment/growth opportunities.
Another option may be a strategy role for a major brand, although in this case you may need to add a good referral to get an interview. In some countries, MBB is also starting to accept people from digital world (people with startups experience, former country managers for e-commerce, etc) for their digital practice.
You may also consider additional jobs allowing to develop the following skills:
Anything that is not allowing you to develop the skills in the previous list would make more difficult for you to receive an invitation. However, even for "unrelated" jobs, you may still have the opportunity to be invited if you would join an MBA program after some years and then reapply.
The ideal company to start with may also be different according to the geography you are applying for. There is a simple method to verify which may be the best profile in a specific country, that is, looking at the profile of people on LinkedIn currently in MBB for your target office and check what they did before. That should give you a more complete overview.