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It's possible to be approved by Chase or Citi without any preapprovals showing up on their site, but if you were declined by Discover and only got $1,000 from Capital One, you shouldn't waste your time -- or, more importantly, the hard pulls -- with Chase or Citi right now.
If your student loans really were in forbearance, you should work on getting those 90-day lates removed or reduced before applying for more cards. It's hard to get good cards from good issuers with relatively recent serious delinquencies and mid-600 scores. (Just my $0.02.)
I'd have to agree with the PP - unfortunately, your file just isn't ready for strong approval chances from prime lenders. Especially Chase, who is quite stringent generally.
Capital One's Platinum is a good place to start. They do CLI every 6 months, if your score/income/payment habits support it. This is one of their tier-2 "rebuild" cards, and it can grow with you as well as establish a history with them that can be useful later to PC to a Quicksilver or get approved for a Venture or whatever else from them you may eventually want and be ready for.
Citi is probably a good one to put on your list for 12 months or so from now, assuming you continue with your work on improving your score. They will eventually start sending you mail offers on their cards; when that happens, you are in their "pre-qual" range. Of course, you may still not succeed until your derogs are resolved, but they are a bit more lenient than Chase.
Chase is going to be something to make a long-term goal. If you keep a clean file for 18-24 months, minimal inquiries, less than 5 new accounts, etc, in that time and your income supports their product requirements, then you'll get in. It took me 3 years to crack Chase - only 1 to crack Amex and Discover - and I've had a Chase checking account for over 20 years and paid off (early every time) three auto loans with them. They just have pretty solid requirements, very low flexibility, and so seem stricter than other lenders. As a side note, I did not have a prequal when I applied for them; my banker did it to see if I could get in on a bonus offer.
I have no insight on Chase because I burned them awhile back and i"m clearly on whatever internal list of NO they keep, but Citi...I have two cards with respectable SL and the lowest APRs offered at the time I apped. Late last year/early this year my scores were in the 500s and I was in student loan default rehab (long story not worth telling) and I'd had one or two missed payments last year due to me just being an idiot and not keeping an eye on things when I was stressed out. I got out of default rehab, focused on at least paying something more than the minimum when I couldn't PIF on my two rebuilder cards, got my scores into the mid 600s and decided to take a chance on apping. I'd gotten one mailer from Citi for the DoubleCash but it seemed like a generic offer and when I checked the prequal on the site nothing ever came up. I got brave one night after getting approved for an AMEX at $1000 and a Discover at $1800 and apped for the Citi DC. Instantly approved @ $3900 Few weeks later apped for the Citi Hilton approved at $5000 and I still wasn't getting prequals on their site. Are those the huge limits of many folks around here? No, but they're respectable and a huge leap from where I was at the beginning of this year. All that to say, give it some time, set a pattern of good credit behavior over the next year or so and then try to app with the big lenders. It can be done with Citi if you time it well. My scores are in the high 600s and low 700s after picking up some new cards that increased my inquiries and dropped my AAoA.