I Samuel 20:27 says:
וַיְהִ֗י מִֽמָּחֳרַ֤ת הַחֹ֙דֶשׁ֙ הַשֵּׁנִ֔י וַיִּפָּקֵ֖ד מְק֣וֹם דָּוִ֑ד (ס) וַיֹּ֤אמֶר שָׁאוּל֙ אֶל־יְהוֹנָתָ֣ן בְּנ֔וֹ מַדּ֜וּעַ לֹא־בָ֧א בֶן־יִשַׁ֛י גַּם־תְּמ֥וֹל גַּם־הַיּ֖וֹם אֶל־הַלָּֽחֶם׃
And it came to pass on the morrow after the new moon, which was the second day, that David’s place was empty; and Saul said unto Jonathan his son: ‘Wherefore cometh not the son of Jesse to the meal, neither yesterday, nor to-day?’
Rashi on I Samuel 20:27:1-2:
ממחרת החדש . ממחרת חידוש הלבנה : השני . ביום שני לחדש :
My translation of above - On the morrow of the New Moon which is the 2nd day of the month.
Metzudat David commentary concurs wuth Rash"i's explanation.
My question is how these 2 commentators know that this is the translation of the bolded words in the verse? In particular, the trope (cantillation) for מִֽמָּחֳרַ֤ת הַחֹ֙דֶשׁ֙ הַשֵּׁנִ֔י seems to make the word הַשֵּׁנִ֔י a modifier of the word הַחֹ֙דֶשׁ֙. Thus, the translation seems that it would mean "The day after the 2nd month", meaning the 2nd day of Chodesh Iyar.
I'm making this assumption of the trope translation by comparing this verse to
בַּחֹ֙דֶשׁ֙ הַשְּׁלִישִׁ֔י לְצֵ֥את בְּנֵי־יִשְׂרָאֵ֖ל מֵאֶ֣רֶץ מִצְרָ֑יִם בַּיּ֣וֹם הַזֶּ֔ה בָּ֖אוּ מִדְבַּ֥ר סִינָֽי׃
In the third month after the children of Israel were gone forth out of the land of Egypt, the same day came they into the wilderness of Sinai.
Similar words and trope, here, but in this case, it's translated as the third month.
Usually, Rash"i explains pshat or provides a source for his explanation. Here, his explanation does not seem to follow pshat, unless there's something evident that I'm not seeing. I'm curious if anyone provided further support or had the same or similar explanation as mine.