Amex frequently issues charge cards to peeps with an EX score in the 657 range. The real question is how much did it drop, and why? If the drop was solely caused by new inquiries, how many new inquiries are we talking about?
I don't think you have anything to worry about.
10 points doesn't seem like much. Though if you have a lot of new inquires, Amex and all CCC might take notice..
Your fine. Don't stress. Your scores will go back up in time. Relax and enjoy life.
My question is when are you in danger of Amex potentially closing your account because they see you as a credit threat?
We don't know the specifics of that question for any creditor out there. All we can say is the general statement that if you exceed a creditor's risk tolerance that they can take adverse action which isn't just closing an account but decreasing credit limit, balance chasing, etc.
Don't just fixate on score. Focus on the data in your reports and be aware of the usual major red flags which are lates and other derogs, high revolving utilization, a lot of credit seeking activity (varies depending on the strength of credit profile). etc.
To add to what others are saying ... Don't worry about the Amex at this point, you're approved and in.... However I would Garden for while and let what you have grown.
Continuing to app and open additional TLs will spook Amex eventually... Amex like to SP new accounts very often.
If American Express closes your account simply because your score on one credit report dropped to 657 they are doing you a favor.
You should look at it as, "Why would I want to do business with a lender who is so scared to lend?"
If American Express is so worried about you defaulting on a card that costs $450 a year and requires the balance to be paid in full every month, then that's their problem, not yours. As soon as they kick you out and your scores go back up, you can go right over to Chase and apply for the Sapphire Reserve which has better benefits and wider acceptance, especially globally. I wouldn't stress over it.