US Law School used to be THE place to prepare for almost any kind of career. Now, the best prep for an international policy career is an MA in IR, Intl Studies, and/or Area Studies. In general, the best places to do such a degree is in DC or NYC but there are important exceptions based on access to internships and policymakers, and for area studies, the region. Other considerations are cost - for example, the DC schools use their MA programs as money makers and so, they are expensive and, generally, don't offer fellowships. A key issue here is to look for a "Professional Masters" program in contrast to a traditional "academic" masters. Thesis option should be available but it is not critical - there is always a substantial research requirement in good programs but it may not be a thesis.
Now, for the PhD. If you only wanted to teach, then moving to a direct PhD would be the way to go from the strongest academic program you could get into. There is also fellowship money there. But that doesn't appear to be your ambition.
For an academic career (in the US) the PhD is best taken in a discipline like Pol Sci or History, etc. For a policy wonk, however, any acredited, decent PhD will do. Interdisciplinary is not only fine but may well be more useful. However, getting the PhD is not something that is particularly useful for getting started in the policy field; it will be far more useful later on.
A last thought: policy wonks are in high demand as adjunct professors at many US schools (as long as they bring a PhD and/or real policy experience) which is a great way to keep a hand in the teaching business while doing a policy day job.

Good luck

JohnT