Even though the other company lost you a month ago, you have absolutely nothing to lose by contacting them now. They obviously believed that you possess the knowledge, experience, and skills to handle the job, and you presumably have not lost any of these qualities in the past month. If the position is still open, you may still have a shot.
It does not sound like you did anything to generate ill will when you accepted the offer from your current employer. They simply came in with the first offer. In fact, it is possible that the company that now interests you has a slow hiring process, and management at that company may actually believe that this process caused them to lose you in the first place.
You should call the individual who offered you the job, indicate that your new position is not working out, and ask if the position is still available. If it is, try to set an appointment. When all is said and done, the worst case is that you move on; the best case is that you move in.
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