Finish your first Ph.D completely. No university allows a student to enroll simultaneously in two programs. If you are working on your thesis and would like to begin thinking about another graduate program, you must wait until you successfully receive your Ph.D from your first program.
Do not overlap Ph.Ds. Once you have a Ph.D in mathematics, you cannot enroll in another program that is similar or equivalent to mathematics. For example, you may enroll in an Engineering Ph.D at a school, but not a Mathematics and Statistics program if you already have a Mathematics Ph.D.
Inquire if the program you are applying for allows graduate students who already have a second Ph.D. As noted in a University of California-Davis Graduate Council letter, it is up to individual departments to enroll graduate students in their programs. (See References.) Therefore, if a student already has a Ph.D, then the new program may say no to that student.
Utilize your connections from your first program to enter into a second program. Graduate school is all about networking. Ask your professors or fellow graduate students if they can help network you and your work to different schools and departments across the country.