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I suppose that if you want a second PhD in Math you have to hide the fact that you already have one. But I do not recommend you to do this. What will you do with your existing publications ? In my university at least, it is a policy not to accept to the PhD program those who already have a PhD. (I don't know whether this policy is written or unwritten. And how strict it is. Probably they do not ask explicitly. And I know people who actually had a PhD (in another country) and who were accepted to our PhD program. They did not mention in their CV that they had a PhD.

It is probably true that most people employed by the "second-tier" math departments are from "top" universities. But it is not true that the hiring committees do not consider other applications. I know many people in "top" universities who have PhD from little-known places.