If you have applied for a lot of cards you may want to take a break (at least 6 months) as recent credit card applications put a ding in your credit score.

If you have other Amex credit cards (not charge cards like the Platinum Card) you may be able to arrange to carve out some of the credit line on your existing card and apply it to a new card--Amex will let you redistribute credit lines among multiple personal credit cards freely, and might let you redistribute to a new account. Not sure how best to go about this but I would imagine that there is a reconsideration desk where you can protest being declined, and they might be able to strike such a deal. But again if you've been rejected multiple times best to wait a bit for your credit profile to recover.