To begin his supplication with praise to Allah almighty and saying the prayer of blessings upon the Prophet PBUH.
The hadith of Faddaalah bin Obaid where he says that "The Messenger of Allah heard a person supplicating during prayer. He did not mention the greatness of Allah, nor did he invoke blessings on the Prophet. The Messenger of Allah said: He made haste. He then called him and said either to him or to any other person: If any of you prays, he should mention the exaltation of his Lord in the beginning and praise Him; he should then invoke blessings on the Prophet; thereafter he should supplicate Allah for anything he wishes".
Therefore, if supplicant begins his supplication with praising Allah almighty, especially with the Quranic verses that praises Allah almighty and then saying the prayer of blessings upon the Prophet, he would have been closer to be answered.