When is it legislated to raise the hands in supplication, and is it permissible to do so between the Adhan and Iqamah, or is it better to supplicate during prayer, and is there a book that explains how the Messenger of Allah, peace and blessings be upon him, used to supplicate?

Raising the hands in supplication (Du'a') is a Sunnah in general and is among the causes for its acceptance, except for supplication within prayer and the supplication of the Khateeb (preacher), where raising the hands is not prescribed. Supplication between the Adhan and Iqamah is prescribed, along with raising the hands. As for supplication during prayer (such as in prostration and before the Tasleem), raising the hands is not prescribed therein. The manner of the Prophet's (peace be upon him) supplication is found in the books of Sunnah, and it has been reported through multiple narrations (Tawatur) that he raised his hands in supplication even in instances other than the prayer for rain (Istisqa').