Our Gemara on Amud Beis uses a phrase, שְׁלֹשָׁה עָשָׂר זְמַן קְהִילָּה לַכֹּל הִיא. The thirteenth of Adar is a time of assembly for all, as the verse indicates, it was on that day that the Jews assembled to fight their enemies, and the main miracle was performed on that day. Consequently, there is no need for a special scriptural derivation to include it as a day that is fit for reading the Megilla.
The Likkutei Halakhos (Tzizis 6:11) notes a recurrent pattern of words in relation to the Megillah and the miracle of Purim. The word “gather”, in various forms repeats itself with unusual regularity. Esther says, לך כנוס כל היהודים go, gather all the Jews (4:16), and several times in the narrative we are told, נקהלו and ויקהלו that the Jews gathered. While Haman disparagingly described the Jews as (3:8) “There is a certain people, scattered and dispersed among the other peoples in all the provinces of your realm.” Scattered is the opposite of gathered.
This word pattern of gathering, and its opposite being scattered, are the Megillah’s way of hinting at a central idea. When the Jews gather together as one, they position themselves to receive miracles, and when they are divided, they are vulnerable. This was latent in Haman’s words, that the Jews are scattered and dispersed, as well as remedied in the gathering of the Jews. It also is notable that when Mordechai heard the news of the impending genocide, he did not fast, pray or lament privately, which is what you might expect of someone who learns terrible news. Instead (Esther 4:1) Mordechai “went through the city, crying out loudly and bitterly”, which might be as if to say, he gathered everyone together.
A gathered and connected community, and even a gathered and connected family, can merit miracles that those who are disconnected may not merit.
Translations Courtesy of Sefaria, except when, sometimes, I disagree with the translation 
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