[subject] + 重新 + [verb phrase]
[subject] + 重新 + [verb] + 一遍 / 一次
重新 goes before a verb and says the action is done over again from the beginning. Usually the first attempt went wrong or is being replaced, so the second time is different.
It often matches English "re-": 重新考虑, "reconsider"; 重新安排, "rearrange".
再 just means the action happens one more time. 重新 means starting over, replacing what was done before. A simple repeat takes 再.
重新 can't stand on its own or follow the verb. It always comes before the verb it modifies.