I don't know details regarding specific allotments as I no longer work there and, generally speaking, we were not privvy to the information much in advance of cardholders. But, let me give you all of my tips.

1. Sign up for American Express' Entertainment Access Newletter, which will come every Wednesday evening, and let you know about any American Express exclusive events or tickets in your are and around the country. Sign up here (sorry can't post links): online.americanexpress.com/entertainmentaccess/home.do#/newsletter.Being on these mailing lists is the most consistent way to get notification of American Express sponsored events. We gave notification for Hamilton, Museum of Ice Cream, Ed Sheeran, Taylor Swift, Drake etc.
2. Sign up for Broadway Across America Amex eClub Newsletter. Here is Houston's: houston.broadway.com/american-express-eclub/. This one varies city-to-city, but will basically get you advanced access to certain braodway shows being scheduled outside of New York, including possibly Hamilton.
3. Your "Promo Code" for 99% of AmericanExpress sponsored events will by the 800 number on the back of your card. It has to be entered without the (1) or spaces, so 800XXXXXXX. So, for example, if you go on to AXS and notice that Elton John and Celine Dion's Vegas performances have an "Amex Exclusive" section, you simply need to enter those 10 digits to see the selection of tickets held back. These tickets are always face value, but kept behind a password and Amex paywall for the exclusive access of cardholders.**It's important to note that, while AmEx has parnerships with AXS and TicketMaster, that obviously does not mean every event that use those ticketing systems are also American Express sponsored. Meaning, you will always see the AmEx logo somewhere on those two sites, but that doesn't mean we necessarily have tickets.
4. As soon as you are aware of events sponsored by AmEX in which you are interested, call Concierge to set-up a pre-order. You set parameters (price, seating section), and they will attempt to use 3 separate inventories (Concierge Exclusive, AmEx exclusive, General Public if applicable) at the time of the on-sale.
5. If it's not an American Express sponsored event, you can call them up to 15 minutes before the general on-sale, and they will wait with you in the virtual queue, using two separate browsers with priority IP access (depending on the website) to essentially triple your chances of success. As I outlined earlier, this also adds a level of pragmatism you would not have if you were to try to do this by yourself. Most high-volume onsales don't allow a map view, but simply force you to choose out of a very wide pricing section. The value difference in the tickets you pull can be drastic. But, when only one person is doing this, you are forced to release tickets without the guarantee of pulling a superior set, or at all. With concierge, you can compare, then release and repull. They are experts and know exactly how much time each "screen" will allow, and how long information takes to fill out, thus they can buy as much time as possible at key moments.

P.S. This is the type of information and services you should be provided with if you actually call concierge and the agent is good at their job.