as mentioned by Frank, it would be possible for you to join a consulting firm with your background, although it would likely be more challenging than with a business/engineering background. The degree to which that would be challenging would depend on the country you are in: while in some countries like Italy, although feasible, it would be pretty hard to get invitations with a background in humanities, in others, like the UK and the US, that would not be a major issue if complemented with other elements.
As for your questions on what you can do to increase your chances, this is what consultants usually look in applications, in order of importance:
- School reputation
- Academic performance
- Work experience
- Extracurricular activities
- Experience abroad
Given that you cannot change your school reputation, you may work on the remaining points, that is:
- Degree: add a minor in business/economics, as you suggested
- Academic performance: maximize your GPA
- Work experience: find internships in business/strategy related fields, if possible with big brands
- Extracurricular activities: start activities in your universities, in particular in clubs related to consulting or activities that would involve interaction with C-levels (eg conferences organization with relevant guests and/or fundraising)
- Experience abroad: join an exchange programme
A final push may be given by a good referral in your desired firm, coming either from Alumni or networking during consulting events.
As for the master, the one that could really make a difference would be an MBA after some years of working experience, in case you cannot join a consulting firm after graduation. After a top MBA, indeed, you should be able to be considered from every top consulting firm, even without previous business/engineering background (I personally helped a former musician to join McKinsey after a top 10 MBA few months ago). Other type of masters could of course also be beneficial, but probably not enough to justify the time and money spent for them, at least if you are applying in the UK and manage to excel in the previous listed areas.