You probably do technically have to file a New York nonresident tax return. @Bsch4477 omitted a critical parenthetical note from the Form IT-203 instructions. What it says is "You have income from a New York source . . . and your New York AGI (Federal amount column) is more than your New York State standard deduction. Complete Form IT-203, lines 1 through 31 and compare the line 31 Federal amount to your standard deduction from the table. . . ." [bold added]
New York AGI in the Federal amount column does not mean only the income you earned in New York. It's essentially the same as your federal AGI, with some possible adjustments for things that New York treats differently from federal. If you worked all year, your federal AGI is surely more than the New York standard deduction. (And the New York standard deduction is much lower than the federal standard deduction.)
But I would not go to the trouble of filing a New York tax return for a matter of one day. Unless you're extraordinarily highly paid, your New York tax return is going to show that you have to pay very little New York tax, or maybe none at all. However, if your W-2 has a state line with NY in box 15 and any New York income in box 16 or New York tax withheld in box 17, then you do have to file a New York nonresident tax return.
