Handicap for PostDoc and Academia Jobs from having a PhD from low ranked universities I am located in the U.S. and the field in question is pure mathematics.

I am wondering, as a doctorata degree holder at a lower tier institution (AMS Group III), how much of a disadvantage will I be at when trying to obtain a postdoc and academia tenure track jobs at say a Group I/II school after graduation? How realistic is it to have such goals and are there anyone out there that has accomplish such feat?

If all hopes are not lost, are there anything I can do to improve my chances down the road (aka polish my CV)? Otherwise, should I drop out and reapply to another program?

Thanks.
I can only answer for my own job market (UK) and field (biology), but in general, what a good pedigree gives you, is second chances. If I was hiring a postdoc, I might take a chance on someone graduating from a good lab at a good uni without a publication under their belt, while from a less well regarded uni, or a research group I'd never heard of, I'd definitely want to see evidence of the ability to do publication quality research.