Patience. That's what I'd recommend. A credit score of 600 is borderline "poor." I don't know how you got there, but as you're clearly aware, there's plenty of room for improvement. Qualifying for a lower mortgage interest rate is a wise goal, but I'm guessing you're at least six to 12 months away from being able to do that, assuming you don't have financial problems in the meantime. Another credit card may not even be the best option for you at this point.
The first step is to make sure your financial house is in order. Looking back over the last six months or so:
- Do you have steady income?
- If you have a mortgage, are you current? If you rent, do you pay on time each month?
- If you're carrying a balance on your current card, are you paying more than the minimum each month?
- Are you able to pay your utilities and other regular bills on time?
If you answered "no" anywhere, do what you need to do to make it a "yes" before applying for another credit card. The last thing you want is to get further in debt and have your score drop even more. If you can answer "yes" to all of those questions based on the last six months or so, you're doing everything right as far as rebuilding your credit history.
And if so, you're in a good position to seek a higher credit limit, which as you probably know will reduce your credit utilization (the amount of available credit you're using) and thereby help you appear to potential lenders as a lower risk. But as I mentioned, it will take time for your credit score to reflect this, and you'll have to decide whether you're able to resist the temptation to dip into that new line of credit during that time.
If you do apply for another credit card and you get rejected, the good news is that your score probably won't drop by much. Your best bet is probably a no frills, no annual fee card.
However this turns out, good luck. Being able to secure a low interest mortgage is probably one of the best reasons out there for doing your best to keep your credit score as high as possible.