Any chance you received - either via e-mail or phone call - a feedback for your rejection?
For instance, if I would have received a feedback like "You are not enough structured in the cases resolution", I believe they would expect me to prepare more than 3 months before applying. I believe the answer could be in your feedback.
If you didn't receive any, I would directly write to the person who sent you the e-mail/person you talked to during the selection process and directly ask to them if you could re-apply (and maybe a feedback to do better next time) or there is a cool-down period. I would go with "I truly care about this role, I believe is the right one for me, ..." and so on, just to let them know you are really committed.