The Biblical new year in the Bible has always been the first month of the year (Nisan, which contains the Passover). This has not changed. The month of Tishrei (containing Rosh Hashanah and the Day of Atonement) is considered the beginning of the civil new year.
In the Bible, God said the beginning of the new year is Nisan (Exodus 12:2). This is now considered the beginning of the religious or ecclesiastical new year.
The building and subsequent destruction of the Second Temple did not alter the calendar structure or the naming of the months. Rather, it facilitated a transformation on how Passover was observed (shifting from Temple sacrifices to home-centered Seder meals) and how the Day of Atonement was understood. The two systems of accounting (religious months starting with Nisan vs. civil years starting with Tishrei) have been part of the Jewish tradition since the Babylonian Exile.
According to rabbinic tradition, the celebration of something new (symbolized by the blowing of trumpets), repentance, atonement, and judgment during Tishrei became the civil new year.
So in Israel today, the new year is the civil year that is celebrated during Tishrei. But the religious first month containing the Passover still exists.
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We can see Jesus in both the Passover and Day of Atonement.
During the original Passover, the blood of a lamb was placed on doorposts so that the Israelites were delivered from a plague of death.
Two thousand years ago, Jesus came to earth and died on the cross for us during Passover (Luke 22-23). Jesus is the Lamb of God that takes away the sins of the world (John 1:29). The blood of the Lamb of God (Jesus' blood) delivers anyone who believes in Him from slavery to sin and spiritual death.
"But He was pierced through for our transgressions,
He was crushed for our iniquities;
The chastening for our well-being fell upon Him,
And by His scourging we are healed." (Isaiah 53:5)
Similarly, the Jewish people were commanded to afflict their souls or confess their sins for hours every year on the Day of Atonement.
When Jesus the Messiah comes again, those who pierced Him will mourn and afflict their souls.
The prophet Zechariah prophesied:
"I will pour out on the house of David and on the inhabitants of Jerusalem, the Spirit of grace and of supplication, so that they will look on Me whom they have pierced; and they will mourn for Him, as one mourns for an only son, and they will weep bitterly over Him like the bitter weeping over a firstborn. In that day there will be great mourning in Jerusalem, like the mourning of Hadadrimmon in the plain of Megiddo." (Zechariah 12:10-11)
In the future, the Jewish people will repent from mostly not recognizing the Messiah the first time He came. And all the survivors of Israel who remain will be saved (Romans 11:26).