The auto-CLI usually occurs 2-3 days after the statement which qualifies you for it is cut. But if you dig into the Terms & Conditions of the credit steps program you'd see that Amex gives itself up to a month after that statement is cut to make the CLI but in practice its only a few days. (See the second condition).
And I would advise you to wait until you get the auto-CLI before applying for any more credit.
@dmjohnsonwrote:Honestly, I would see if you prequalify or the Delta Gold card. (I know checking prequal offers from them is hard when already a member.) I was approved in January for that card, my first card with AMEX, with a roughly 640 score for a $1k limit. The offer I was given was $35,000 miles for $1k spend in 3 months, plus no AF for the first year. I know it has a $95 AF, but that can be taken care of depending on how often you fly delta, and to where. with the free checked bag, 2 fround trip flights in a year will negate the AF anyway.
I do not know what the gold offer looks like now, and I understand if you do not like the AF idea, this was just my experience. Either way, it is a nice feeling to have a card that will reward you with miles, especially if you like to travel! I spent most of my very few adult years off racing sprint cars every weekend, so now that I am kind of taking a break from that, we are trying to travel as much as we can!
Good Luck!
It shows that I'm preapproved for the Delta Blue and also the Delta Gold and Delta Platinum and Reserve, among other cards. Yes, I looked at the Delta Gold details but I don't want a card with an annual fee though. I've never flown before in my life. I travel occasionally but not often, and I'm terrified of flying but if I want to get anywhere faster than driving a car I'll need to overcome my fear of flying. So this summer I'll be flying at least once. The Delta Gold free checked bag wouldn't be very useful to me since I don't plan on flying often. Thank you for the data points though 
I feel more confident that I'd be approved for the Delta Blue.
I'll try to hold off until my statement cuts and I see an auto CLI and then apply for the Delta Blue. It's difficult to wait, but I guess it might be worth it.
@Anonymous Right now the signup bonus on the Gold is fantastic, and the AF is waived the first year. Trust me, "go for the gold," and then if you don't want the AF after the free year is up you can call or chat and product change down to the no-AF blue one. The signup bonus alone is enough for a free round-trip Main Cabin ticket to Europe or multiple round-trip domestic Main Cabin tickets! Plus there's a $50 statement credit with your first Delta purchase (suggestion: buy a $50 Delta gift card using a desktop browser). I very seriously doubt that the underwriting criteria is different among the Delta cards.
The AF will be charged on your anniversary date. You have 30 days to have it waived or PC. With the recent Amex crackdown on bonus abuse, I’d play it safe and not cancel or PC until the one year mark.