I called back, got the machine again. I KNOW they do now have caller ID so I just hung up. I'm not giving up!
Would it be ok to leave another message tomorrow that is IF I get the machine again? Or should I wait a day between messages but continue to try to call a few times a day?

Yes, the old saying is true, Don't wait for them to call YOU.
I know that I just don't want to be TOO pushy.