Yes, there is an Amex reconsideration line for applicants to call and request a second review of their denied American Express credit card application. The Amex reconsideration line is (800) 567-1083. Amex credit analysts have the authority to overturn a denied application over the phone, but there's no guarantee they'll reverse the original decision.
Before requesting an Amex reconsideration, make sure you have a valid reason why your application deserves a second look. For example, there may have been a significant positive change in your financial situation or your credit report since you applied. Be prepared to provide any supporting documentation, if requested. Feel free to also take a look at WalletHub's credit card reconsideration guide before requesting reconsideration.
Alternatives for denied applicants
If your Amex reconsideration is unsuccessful, you still have other options. The obvious choice would be to apply for another card that might be easier to get. American Express cards require good credit or better (a credit score of 700+), so if your credit standing is holding you back, you'll have to either seek out a card from another issuer or sign on to someone else's Amex account as an authorized user.
If you prefer, you can simply work on remedying the issue(s) that resulted in the denied application in the first place, and apply for that same Amex card again. American Express does recommend waiting at least 30 days from the date you received the denial letter before submitting a second application, but most improvements will take longer than that anyway. Feel free to also check out WalletHub's guide on how to improve your credit score.