If you have been declined then you can ask the credit card company for a copy of your credit score to see why (not sure how quickly you need to do this though). This can give you an idea of what you might need to work on to get your credit score up for a better CC offer.

The other thing is to make sure you are paying your CC bill on time and do what you can to pay more than the minimum. Paying on time is huge for your credit and can go a long way to getting you a better offer.

Lastly, don't add any more debt to your cards. The more credit you are using the worse your credit utilization score will be (and the more interest you will be paying).

If you have been good with your payments but just missed once I would call the CC company and ask them if they could put your rate back to where it was. Explain how it was a one-time incident and you are working hard to pay off your balance. Tell them you love using their card and want to keep using it as you build their credit but you have been getting other offers.