Network's impact on lateraling

Majority on this forum would agree that having a strong network is influential for a successful upstream lateral.

I was curious as to how does that impact is actually manifested throughout the process. 

On a surface level, it obviously would seem to depend on the seniority of the person referring, but I am moreso curious about how much of an impact does a referral really have if, say, a less senior person, like an analyst II/ASO pushes your resume to HR given your background is still relatively weak compared to a typical profile of a candidate for that bank.

My context: come from a target, work in a small regional boutique in a big city, want to lateral to larger upper MM/EB/BB over the next 6ish months; would like to cement realistic expectations when it comes to leveraging my existing network (macro aside).

Thanks