Hard to really help without seeing what your resume actually looks like (and we can help you optimize said resume anyway). Having said that:
1. If you have solid pro experience + great Masters GPA, your undergrad GPA will be a little less important - especially if a long time ago. Note I said "a little less", not "completely irrelevant": we use formulas to decide who comes to an interview, and there's a place for undergrade GPA - but since there's also a place for Masters GPA, the undergrad GPA mathematically becomes less important
2. If your undergrad GPA is low due to hard grading standards, you can also mention that. Do you have a comparatively better ranking for example? Your question suggests this may not be the case, just thought I'd mention anyway
3. LinkedIn shouldn't be over-emphasized. Use it to find people that you could reach out to, but it is only 'a' tool, not 'the truth'. Focus on finding consultants who have a similar background to yours
4. Cover letter should absolutely explain your story a little bit, even if it probably won't be read (we had a good thread on this about a week ago)
5. Assuming for a second that you do not have the right profile for MBB for example. Don't forget to look at Tier 2 consultancies, who may be less demanding in terms of grades; don't forget also you can start in industry (or consulting) for a few years before trying again. All these options will help you minimize the emphasis placed on your grades
6. Last but not least - I had poor grades in my first degree (a master's degree, I literally had one of the lowest GPAs in my entire class), yet eventually found my way into BCG and have done mighty well since then (not bragging, just fact + showing how stup*d I was in my early 20s). Grades are just one metric we use to decide who gets in and who does not. Sometimes, people "wake up", like I did. Don't let a disappointing performance when you were 18-22 dictate your life for the next 40 years.