The period up to the destruction of the Temple by the Romans is called the Second Temple Period. But in the New Testament they refer to Herod the Great's having built the Temple. Did he just restore and improve the one built after the return from the Babylonian captivity?
Herod made it grand
He had something to prove - after all, Herod was the son of Edomites -- as an aside, John Hyrcanus, one of the Maccabean rulers of Judea conquered the Edomites around 100 BC and forcibly converted them to Judaism.
Herod didn't "just" restore and improve - he improved it incredibly, a complete rebuild, akin to the rebuilding of St. Peter's cathedral in the Vatican
Modern-Day Judaism as we know it is descended from the Pharisee sect of 2nd temple Judaism and it arose in the destruction of the 2nd temple in 69 AD when Rabbi Ben Zakkai re-formed the religion away from the temple and priests to synagogues and rabbis (refer "First century Judaism in Crisis" by Neusner)
Christianity is descended from another 2nd temple Judaism sect, the Jesus-movement or the Messianic Judaism (distinct from modern day Messianic Jews who are Pharisee-descended modern Jews who have become Christians)
The modern day Jews are our brothers and it is a sister religion to us - that explains a lot of the early animosity between Christians and modern-day (CE) Jews is around these two rival views -- CE Judaism is in fact 37 years younger than Christianity - since it starts in the Council of Jamnia in 70 AD that settles the Jewish Torah and the CE J decide to not follow the Septuagint as that is used by the Jesus-movement Jews to "prove" their point, while the Jesus-movement Jews take pains to stick away from the term "Jew" due to the bad connotations after the Jewish-Roman wars