"On this day they came to the wilderness of Sinai" (Exodus 19:1). It was the New Moon. For this reason it is said, "On this day, in the third month."
Israel Reaches the Wilderness of Sinai on the New Moon
Yalkut Shimoni on Torah 274:2
בַּיּוֹם הַזֶּה בָּאוּ מִדְבַּר סִינָי, רֹאשׁ חֹדֶשׁ הָיָה, לְכָךְ נֶאֱמַר בַּיּוֹם הַזֶּה בַּחֹדֶשׁ הַשְּׁלִישִׁי.