First, you need to define what kind of consulting you want to be part of. It is apparently not "IT Consulting" (which is consulting in name only, but you still have to clarify). Are you looking to join MBB? Tier 2? Boutiques?
Let's assume MBB for now, for argument's sake. Companies aren't going to put much weight on your actual knowledge, but you definitely have to show a career progression. Moving around every few months / year is a red flag since we don't know how well you performed or if you just couldn't work with the people around you
For boutiques, progression may be a little less important - but you will still need to show you can stay in a role for more than a year: it takes a long time and a lot of money to train a consultant, they aren't going to want to do this if you plan to move on a year later.
I understand where you come from, but think you may need to sit still for a little bit, and get promoted to show you are appreciated. Then your profile will look much stronger