I'm glad to hear that you won't carry a balance, and even happier to hear that you're planning to re-establish your good credit. It's all too easy to get discouraged and figure, "Why bother?"
And two good reasons to bother are that first, credit scores tend to go up if you're demonstrating good financial habits, and second, with a lousy credit score, you're pretty much doomed to always pay more interest on any loan, whether on a credit card, a mortgage or an auto loan.
So you have a credit score of 650. That isn't terrible--but as you know, it's not exactly considered something to write home about either. That said, some credit card issuers consider 650 good or reasonably decent, and so you should be able to get a credit card without too much difficulty. It won't, however, be a card with a low purchase APR, but you already know that.
I don't know--nor does anyone else in the credit card and personal finance industry--any of the trade secrets that credit card issuers use when determining who gets a credit card and who doesn't.
FICO scores aren't everything, of course. If you still have trouble getting approved, there may be other things in your credit history that are keeping you from getting a credit card with decent terms. As another option, you might want to consider a department store credit card. With a lot of stores, the bar for getting one of these cards is pretty low.
The main pitfall to getting a department store credit card, especially if it's to take advantage of some in-store discount when you sign up, is that all you need to do is carry over a balance for a month, and the interest could wipe out whatever you hoped to save. Many store credit cards charge exceptionally high interest rates. If you carry a revolving balance, that does you no favors, and if you wind up with a lot of department store credit cards, that can actually hurt your credit score even if you keep them all paid off.
But, you know, if you're looking to build up your credit, and nobody else will take you, one department store credit card may help build your credit to the point that you can eventually get one that offers some modest rewards.
Secured credit cards
And if you're really desperate and simply having no luck getting a credit card issued to you, secured credit cards, where you put some of your own money down into the card, can be a good vehicle for building credit. But with a 650 credit score, you should be past that. Whatever you do, good luck!