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First of all, note that improving your credit score is not a quick process. It may take up to a couple of years to turn bad credit into good credit. But if you want to improve your credit score faster, keep your credit accounts in good standing, keep balances low, and avoid missing payments. When you pay more than the minimum payment, you can significantly increase your credit score. Note that there are 5 main factors that make up a credit score: payment history, credit utilization (amount of debt), length of credit history, new credit, and credit mix. You should always monitor your payments so that to be able to cope with all of them. Secondly, your total outstanding debt shouldn't exceed 30% of your total available credit and on each individual credit card. If you always pay on time and reduce your debt, you may see first changes in your scores after 3-6 months. If these changes are positive, then you're doing everything right. If not, then you need to check your credit reports to see where you could improve.
Note that to qualify for any credit card, you should have no active bankruptcy, no late payments, and no excess of the limit on your current credit accounts. Therefore, we would recommend you pay off all your late payments first and then pay on time at least for two months on all accounts. Only after that you can try to apply for a new card. For example, you may consider the Milestone® Mastercard® - Less Than Perfect Credit Considered with 24.9% APR for purchases and a $35 - $99* annual fee depending on your creditworthiness. With this card, your account history will be reported to three major credit bureaus and therefore you will continue building your credit history.