why the 13th?

Megillat Esther

Haman selected the 13th of Adar as the date to decree the uprising of the people against the Jews.

In 3:7 we find the lots cast and falling on Adar. Nothing is mentioned about any sepcific day in Adar.

In 3:12 we find Ahashverosh signing the decree on the 13th of Nissan and sending out his messengers to announce it.

In 3:13 we find, "וְנִשְׁלוֹחַ סְפָרִים בְּיַד הָרָצִים, אֶל-כָּל-מְדִינוֹת הַמֶּלֶךְ--לְהַשְׁמִיד לַהֲרֹג וּלְאַבֵּד אֶת-כָּל-הַיְּהוּדִים מִנַּעַר וְעַד-זָקֵן טַף וְנָשִׁים בְּיוֹם אֶחָד, בִּשְׁלוֹשָׁה עָשָׂר לְחֹדֶשׁ שְׁנֵים-עָשָׂר הוּא-חֹדֶשׁ אֲדָר; וּשְׁלָלָם, לָבוֹז." - And letters were sent by posts into all the king's provinces, to destroy, to slay, and to cause to perish, all Jews, both young and old, little children and women, in one day, even upon the thirteenth day of the twelfth month, which is the month Adar, and to take the spoil of them for a prey.

Suddenly the 13th of Adar became the day for the extermination of the Jews (notice also, as per my previous post about 7 Adar, that the passuk in which is announced the date of the eventual uprising on the 13th of Nissan is in passuk 13).

How did that day get selected? Was it simply 12 full months from the day the letter was signed? For a suspicious Haman would it not have been better for him to insist on the date selected be the 7th of Adar, since he was so concerned about the bad luck time period of the Jews?