Amanda, consider (if possible) getting a letter from your current employer attesting to your good attendance/punctuality record at your current job. That may sway someone who is thinking 'how do we know she's changed?'.
Also, you may want to consider taking whatever you can get at first. If you're trying to get rehired, it looks better not to be too picky, but to be so eager that you'll take anything just to get back on. Where I used to work, if a person turned down a promotion or special assignment because it wasn't to their specific liking, they were usually never offered anything again. We learned quickly to take what was offered and then work towards our 'ideal' job.