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Your cap 1 will unsecure at 11 months of perfect payment history so that should be a keeper Imo. Have you tried your local credit union? Do you qualify for nfcu? Penfed etc... Dv
Check your preapproval offers to prioritize what card you'd like (rewards, no AF, balance transfer). I had similar scores last month and decided to go for a card that I would use frequently, but also maximize rewards to keep me sane. I would say in your instance, try a capital one "quickilver/ platinum," chase freedom/ freedom unlimited/ slate, or any card that makes sense for you. Try rebuilding first and who knows what you'll get in a month...best of luck!
I was in that situation when I started rebuilding. I started with crappy cards and crappy limits like you and was appreciative of them. My breakthru card was Bank of America with SL of 5k. Things got easier after that. I had a banking relationship with them though. If I had been banking with Chase, I would have done the same with Chase, etc. So if you have not already, focus on where you have a banking relationship OR on banks you have never burned. Sounds like you are off to a great start with what you have. That AMEX will grow and so will the Chase. The answer might be that you are already doing what you should be doing. Sometimes it just comes down to time and patience. Stay on those CLI requests as soon as you are eligible and push those buttons monthly if SPs.
Synchrony Bank is amazing! That might be the correct recommendation actually. I apped for the Synchrony Paypal card and was approved for 3k. 4 months later I was immediately approved with no questions up to 10k thru the live chat.