It's difficult but not impossible. I can share my experience about tech recruiting.
I didn't bother looking at consulting or banking from the get go and started focusing on developing a game plan for tech recruiting. I reached out to 2nd years, alumni I have strong commonalities with, and my own network between September and October. I consider these to be my inner circle or warmer network contacts. I started reaching out beyond that to other alumni and 3rd degree connections between November and December. By the January, when people who flamed out of consulting and banking have joined the fray and are still figuring out the nuances of tech recruiting, I've already racked up 8-10 referrals at several mid to large cap tech firms (funny story, none of the firms I was referred at interviewed me, but I got a number of interviews from several really awesome firms that really surprised me).
As an international student, I lost out on interview opportunities with 3-4 firms because of my non residence status - prior to interviewing me, they asked me about my citizenship and rejected my application as soon as they found out I'm not a resident.
All in all, I came away with interviews with about 10 different firms (ranging from pre-IPO firms to mega cap firms), was given 3 offers, with 2 still in progress but halted because I signed by then. I only interviewed for strategy, business development, and marketing roles - and keep in mind I have none of these experiences coming entirely from a finance background outside of the US.
Most people who came away with something they didn't like or couldn't find a job were international students who went all in on banking or consulting, didn't secure an offer in January, spent too much time regrouping for plan B, and pivoted into the game a bit too late.
Then there's people with unrealistic expectations about jumping into PE and VC without the required experience.
You're going to try for consulting, so I recommend having a backup plan. Talk to current students and see how they planned for it. Also ask them about firms that has Kellogg as a top target school, always good to pay attention to those firms because getting interviews from them might require little to no effort.
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