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I can't stress enough the importance of applying for what you want rather than settling for what you think you might be approved for. If you are interested in bank cards, I would take a look at the various major lenders' prequalification sites to see if anything comes up, specifically with a single APR or a small APR range listed (not "14.49% - 26.49% depending upon creditworthiness" or similar language). Those are generally a sign of better approval odds, or near-certainty with some lenders.
Are there particular stores that you shop regularly where having a cobranded or store card would save you money on purchases you are already making? If so, it may make sense to Google credit pulls database and compare your scores to those that report being approved for the same products.
Finally, do you do much traveling? If so, by what means and where do you stay? A charge card from American Express would be a fairly likely approval with your scores, but you would want to make sure that the annual fee is worth it to you.
An app spree is not a good idea. Find a credit card (just one) that would really add something to your portfolio and apply for that one. Wait six months before going after a second card.
If Chase is part of the plan, you need to do it right. First, wait until your oldest card is a year old. Second, stay away from 5/24. If you really want a Chase card, you could grab a different card now if you'd like, then go for Chase as your next card once your profile is ready.
Some lenders are taking a very dim view of people who aggresively seek credit. I would not be engaging in app sprees and instead just focus on slowly aquiring the cards you want/need. Being too agressive and going for a number of cards at once may make life much more difficult later on.
Great job on improving your credit, OP! Your scores not bad, but I would like some data points so that I can give you a well-informed answer.
Rude, your reply contributes nothing to this conversation, if you have nothing constructive to say, well....