Importance of School Prestige when climbing the ladder

I had recently just got a S&T Internship at a BB and am very happy about it. I come from a non-target in the West Coast and know first hand how hard it can be when you're not a target school level candidate going through recruitment. Although I sort of have my foot in the door for the industry with the internship I also have the opportunity to transfer to either Boston College, Boston University, or Northeastern. I'm debating whether the move is worth it considering the context of the situation, and the one question I always have on my mind is, "Does school really matter when you're already in the firm and trying climb from Analyst, to Associate, to VP, etc. or does school only matter when trying to break into IB from college" I know those schools aren't exactly target schools but have more of a presence in IB than my current institution. Curious to see what you guys think.