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Browse credit cards from a variety of issuers to see if there's a better card for you.
Check out the credit pulls database and search for the card and you'll see the scores people were approved/denied with which will give you a good idea.
As an aside, have you done anything to try and get that late payment removed? If it's your only one and it's only a 30 day, the least severe, you stand a pretty good chance depeding on the lender at getting it removed. If you did, you'd probably push yourself from the mid 600's to a score of about 700 and at that point you'd have to worry far less about denials.
For the Chase, no, and for the Synch it'll be an SP CLI so wait it out, especially if you can get the baddie removed. Congrats and good luck!