From mid-cap to Large-Cap

Good morning everyone,

I'm currently m&a intern in the midcap team of a BB but in a french Regional office. I'm graduated from a non-target school and I'm looking for my first permanent contract in IB. Mid-cap is interesting but I think i would prefer to start my career in a Large Cap team. I applied to mid-cap and Large-cap offers in london and Paris but only mid-cap boutique are interested in my profile.

My question is, how can I get into a Large Cap team from big banks or EB in Paris or London. Every Analyst 1 in banks or EB in paris or London seems to come from Top schools (HEC, ESSEC, LSE, Oxbridge) or to have already experiences in large cap. Do you think it's even possible for me to get a job in a large cap team? What would you recommend me to do to make that type of move?