Three pilgrimage festivals shall you celebrate for Me during the year. You shall observe the Festival of Matzos; seven days shall you eat matzos, as I have commanded you, at the appointed time of the month of springtime, for in it you left Egypt; you shall not be seen before Me empty-handed. And the Festival of the Harvest [Shavuos] of the first fruits of your labor that you sow in the field; and the Festival of Ingathering [Succos] at the close of the year when you gather in your work from the field. (Ex. 20:14-16, Artscroll)And emphasizing that the close of one year is the beginning of another:
You shall count for yourself seven cycles of sabbatical years, seven years seven times; the years of the seven cycles of sabbatical years shall be for you forty-nine years.You shall sound a broken blast on the shofar, in the seventh month, on the tenth of the month; on the Day of Atonement you shall sound the shofar throughout your land. You shall sanctify the fiftieth year and proclaim freedom throughout the land for all of its inhabitants (Leviticus 25:8-10)We see clearly that the Sabbatical and Jubilee years begin in Tishrei. The Navi Yechezkel goes on to call "the tenth day of the month" which is most appropriately identified as Yom Kippur (The Day of Atonement, the only holiday which falls on the 10th of the month and which marks the liberation at the start of the Jubilee year as we just saw) "b'Rosh Hashannah", at the beginning of the year (Ezekiel 40:1).
It is incontestable that Tishrei marks the transition from one year to the other. It is also incontestable that while the Exodus took place in the middle of Nissan, the begging of Nissan is regarded as the "New Year". It is therefore clear that while the month of Tishrei has the character of the start of the year, it is implicit that the specific day to associate as being the "New Year" would be the first day of the month...Yom HaTeruah, which is not given explicit significance in Chumash.