While I don't know for sure what's typical, and YMMV always, I received my 61-day cli on 1 of my 2 revolvers and about a couple of weeks later Amex began to pre-approve me for some of their other cards through their account website. I would start there if I were you, checking every few days or so. That said, although I was pre-approved I was not allowed to apply until I hit the 90-day mark (which is today!) and my application was canceled (not denied) instantly but I was told to call back on day 91.
Congrats on your 2x CLI!
"Cards>Personal Cards>Check for Pre-qualified Offers", see below.
You can also select "Apply Now" for the card you're interested in. Two things will happen in my experience: You get a "Yay, you're pre-approved message" or something vague about "expanding your portfolio". Generally, you want to ignore the second message as it's likely you won't be approved, but YMMV, although I'm not certain yet if the first one is a sure thing. I find out tomorrow!!! 
I know this is old, but for the sake of assisting someone else... OP, you have to click on the apply button to see if you are pre-approved or if you get some other message! 
“Most people work just hard enough not to get fired and get paid just enough not to quit.”
