Switch from Finance to Public Sector?
I'm currently working in finance and have come to realize that I'm not as passionate about this area as I once thought. Although my undergrad is in business/finance, I am now becoming more interested in public policy or working for a think tank/consulting firm that uses stats and publishes research that guide policy decisions (such as how to allocate government budgets for various projects that will lead to higher standards of living, how to reform broken public systems, etc.). While part of this may be driven from the hype surrounding the elections, I've also been thinking for a while that I'm just not a great fit for the high finance world in terms of personality and that my personal interests would be suited to the public sector.
My question, however, is how can I go about making this transition in the future? I have limited work experience and am likely going to spend another 2-3 years in finance to get more work experience, but I'm wondering what I should do for grad school? Based on my research, there seem to be Masters in Public Administration or Masters in Public Policy degrees that cater to these interests, but would I be at a disadvantage coming from a non-quantitative business background from a non-target? I mean, for designing policy, wouldn't someone with a PHD in economics from an ivy be better qualified or do people from MPA/MPP programs get positions such as the one I mentioned?
If I do this, I'll only be aiming for the top schools. I made the mistake of not going to a great undergraduate school and am paying the price for it by being at a dead-end job at a no-name firm that I'm looking to lateral from, but given my newfound interests, does anyone have advice on how I can make this transition? I know this might not be the right forum, but I thought that there might be someone who has experience with this, which is why I thought it'd be worth asking.