When did you graduate? If it's within the last 3 years or so, you may still be able to apply to graduate schemes.
Otherwise, getting more experience in industry, perhaps in your present organisation if it's big/well-known, and if you think you can accelerate up the ranks there, would be an option so that you can move into business process improvement consulting somewhere down the line. You would need to focus on getting experience with as many clients and project approaches as possible.
An MBA, or MA/MSc is another option, although I think this is best applied in addition to the industry experience mentioned above.
The main priority for you should be maintaining momentum in your career and showing continuous learning and progression in areas that will transfer directly into consulting. However, as the other poster suggests, the best thing is to get into consulting sooner rather than later if that is always the best environment to demonstrate consulting skills.