Is it permissible to prostrate oneself to supplicate for a need, other than during prayer, such as prostrating oneself for supplication after the Taslim (the salutation of peace that concludes the prayer)?

There is no objection to prostrating outside of prayer for supplication, due to the permissibility of prostration for a legitimate purpose such as supplication. Sheikh al-Islam Ibn Taymiyyah and others have explicitly stated this. However, it should not be made a habit; rather, a Muslim may do it occasionally. He should also be keen to supplicate during prostration in prayer, as that is more complete and superior.