define great internships. If you already have interned at T1 firms in your preferred practice area then work your ass off to get a callback and convert it into a PPO. Sure there is a lot of factors that go into PPOs that you can't control. But whatever is in your control, at least nail that. If you do not have T1 internships, then the truth is your internships may be great but T1 HR does not give a f. All they see is (in that order - institution + grades, past internships - they basically want prior T1 experience in the same practice area, or at least 2-3 boutique and t2/t3 stints, publications, moots, certifications are good to have. Some firms like NDA or S&R value publications more, for others I don't think they are going to break or make it.) And even if you end it at a T3 is it possible to make lateral moves into T1, but you have have to network smart and be very good at what you do so that people actually want to hire you. A lot of people do it. For lateral hiring there are also consultants like Vahura who may be able to help you out.

Don't just stack internships. Perfect the craft. As Karl JUng said - No matter how isolated you feel you may be, if you do your work truly and conscientiously, unknown allies would come and seek you. Do more internships and decide that you're going to outperformer every intern the firm has ever seen. You'll make the partners want to hire you. Alos work on your oratory skills, personality, dressing sense, it also makes a huge difference in how people perceive you.