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Top 20% at T20, did OK in preselects but there's a few firms I really want to interview with (and some put me on the "alternate" list). School is 100% pre-select, but firms have very few slots. Any advice on how/what kind of approaches are effective? Did a search, advice is mixed.

Why not? They aren't gonna say "This a-hole. I can't believe they emailed us for a job. Ding 'em."

At best, they make some space for you. At worst they ignore you and you're no worse off.

Obv worst they can say is no. More importantly, if you were the last person they dinged, they may have actually liked you but ran out of slots. this means that when more attractive candidates pick other firms, it would make sense to take you now. Not saying this is their thinking, but it's definitely enough to say you should def mail them.

Any suggestions how to phrase something like this. Would you just say in your email to the recruiting contact that you applied but weren't selected for a spot during OCI but you are still interested and would love to come in for an interview if that is possible.

OP I did this and so far it's working. I've been added to three firms' schedules so far, and I was below their cutoff. Only one said no. So do it. Make your case and see. You have nothing to lose.