Depends on where you went, your grades, any work experience, etc., but there's no reason not to target the bigger/more prestigious companies and institutions if you have a solid background.
I mean...you don't really need a banking/finance-specific background to get a good job at a bank - a lot of what they'll look at is skill-based. If you're good with numbers and data analysis, there's a pretty wide variety of opportunities...or you could utilize your skills to be a project manger, etc. To use a personal example, my legal background has opened up doors to more senior roles in compliance and risk management, even though I majored in political science and have no background in actual bank stuff. My manager started as a correctional officer before getting a job at the Fed.