With a credit rating in the high 600's to low 700's you shouldn't have any issues. This is good credit. I wouldn't equate bad credit with not being able to get a credit card. If you have a car loan, you have good credit as long as you've never made a late payment.
Credit cards are evil. They can and will get you in finanical trouble more quickly than anything else.
Some C Cards will look at your bank accounts. If you have little or no money in checking and savings, that will throw a flag. Any late payment will also throw a flag. But these should reflect in your credit score.
If you must have a credit card, go to Sears, JC Penney or Macys and apply for one of their cards. They will usually give you one with a small credit limit, maybe $500. Buy something and make full payment EVERY MONTH that you buy something. Do this a few times within the first year. Don't make a late payment and by this time next year, you will probably be able to have a Visa or Mastercard.
But be forwarned, credit cards are bad, they will get you in trouble very quickly. I swore off credit cards about 10 years ago. I either pay cash for what I want or I utilize the 0% interest many of the stores are advertising.
Buy something on their dime. Believe it or not, this will also build your credit.
Good Luck!
Jim Smith