I agree with item 2 of Alexandre Eremenko's answer. Lower tier departments who care more about how they appear to the dean rather than the actual quality of the research do sometimes go after Ph.D.'s from top schools because it looks better. But departments that have actual aspirations of having strong research will look at anyone who has a strong track record, regardless of where that person got a Ph.D.
Here are some thoughts:
- Get to know as many strong senior mathematicians as possible, especially those who work in your field or closely related ones. Try to make sure they are familiar with your work by giving seminars at their schools, attending and giving talks at conferences, posting your papers on arxiv, and sending a PDF of your paper to people you think might be interested (but don't overdo this).
These people are invaluable in at least two ways: 1) Putting in a good word for you, whether in a formal letter or informally by email or phone. 2) Providing useful guidance to you on which departments are hiring and where you stand the best chance.
Who writes strong letters for you matters at least as much as where you got a Ph.D.
When visiting a department that you think you have a shot at getting a job at, don't be afraid to ask them what the prospects are and what they are looking for. Getting a better sense of how each department chooses how to hire is invaluable.
Maintain good and friendly relations with absolutely everyone in your field. For example, when you write a paper, cite people generously, including anyone who has done anything related to your work, whether you used the work or not in your paper. Showing your appreciation for others' work by citing them is a good way to build a lot of goodwill.