Just block Amex so they can't bother you as otherwise they will call every single day about accounts that don't even have anything due just because the system override let you go over the limit so its flagging your account as one they must call now.
Also, paying by 8PM MST (10PM EST) on the due date is most important in order to prevent late payment penalties and interest, credit score reduction, loss of Amex rewards, and loss of system override. Its better just to pay and let it bounce at the bank once and pay the $36 fee to bank and $36 fee to Amex for the overdraft than to pay late.
We recently ran into this issue where we had due date on a Thursday so we waited until 9:10pm EST to pay it hoping it wouldn't post until the following week as we'd have the money in the bank come that following Monday. The next day on Friday the account was clear and we billed a lot of new charges to the account. Even Monday we rebilled a bunch of additional charges to the account so much that we maxed card $300K above its limit seeing that the system override was in full effect. It wasn't until 2 business days into the following week that we saw the $36 overdraft fee on Amex side but by then we had new posted charges taking the balance $300K above its limit and that was after counting the payment that eventually cleared the following week on the 2nd attempt Amex made. We earned all the rewards and account shows as clear and all we had to pay was the $36 twice.
Based on our experiment/experience, paying on due date by 8PM MST is a must. Amex will reattempt for total of 3 times before it will go down as a no payment. 2x$36 = $72 in Overdraft fees are much better alternative than late payment penalties and interest, credit score reduction, loss of Amex rewards, and loss of system override. When you pay just know it will take about 2 business days for them to find out it bounced so you have all that time to charge the card up some more so if you have vendors that want money, get them to accept the Amex with markup. That way you are postponing that money coming out of your bank. Also, for the new charges, place them the day after statement date so that way you have 29 days until they post as due.
These are all the best tricks to managing Amex calls and card payments.