Pre-Med Engineer to WS

Reposting from another forum because this is probably the best place for it. I realize how unlikely this probably is, but I figured I might as well give it a shot. I'm an engineering major (not EE or CS) at a top engineering school but generally a semi-target for finance/consulting firms. I've done very well academically (3.9+ GPA) and have good ECs, so as you can imagine, that took a small toll on my social life. Don't have any internships in industry, though.

I'm taking a gap year to apply to med school, and while I can do a boring gap year job like research at a university or something, it'd be great to experience industry, specifically finance and consulting. I previously had a intern/externship (not a summer internship) at a major quant hedge fund (Two Sigma, DE Shaw, Citadel, Jane Street) that exposed me to a lot of finance and I thought the pace and content was pretty exciting. Is there any chance I might be able to land a full time position or internship at a mid-high level bank or consulting firm if I am applying now (spring cycle) if I'm a graduating senior? I know most firms generally wrap up during the fall, but I had other priorities I had to focus on.

I know how unlikely it is to get a position like what I'm asking for but I had a friend that got an analyst position at BB with no relevant experience whatsoever (same major and everything), so that inspired me to make this post.