Nonprofit Type Internship to Consulting?

Alright, so I'm a junior in humanities at a target school (Harvard/Yale/Princeton) with a 3.6 GPA. I already know I'm not the most cookie-cutter candidate for good jobs. Unfortunately, I'm not interning in investment banking or a name-brand firm this summer.

I'm working at a history museum as an Academic Research Analyst. Basically, I get to conduct original quantitative research, working with a couple graduate students and PhDs to produce a paper and that will be published in late summer, and also giving some presentations in New York City.

I interview quite well, but I know this is far from being the kind of thing that consulting firms look for. My other internship experience is at a boutique real estate firm, doing work in Excel. During junior year recruitment for internships, I made it pretty far in the process for one MBB, but ultimately struck out because of a lack of preparation.

Anyway, my question is: Could I get interviews for second- or third-tier consulting firms with this experience? What would you recommend I do? I definitely feel like a loser at this point, but I'm hoping I can make something out of this experience and maybe get a decent job.

Thanks.