The High Priest's headdress was to be made of fine linen on which a gold plate was attached:
36 You shall make a frontlet of pure gold and engrave on it the seal inscription: "Holy to the LORD." 37 Suspend it on a cord of blue, so that it may remain on the headdress; it shall remain on the front of the headdress. 38 It shall be on Aaron's forehead, that Aaron may take away any sin arising from the holy things that the Israelites consecrate, from any of their sacred donations; it shall be on his forehead at all times, to win acceptance for them before the LORD. 39 You shall make the fringed tunic of fine linen. You shall make the headdress of fine linen. You shall make the sash of embroidered work. (Exodus 28) [NJPS]
In Antiquities of the Jews, Josephus describes the garments and headdress of the High Priest. He gives a very detailed description of a crown worn above the inscribed gold plate:
172 His head-dress was the same as that described earlier and made like that of all the other priests, but above it was another, embroidered in blue and round it a golden crown, in three rows, from which arose a cup of gold like the herb we call Saccharus, but which is called Hyoscyamus by Greeks who are skilled in the cutting of roots. 173 In case anyone has seen this herb but has not learned its name or is unaware of its nature, or, though knowing the name does not know the herb when he sees it, I shall now describe it. 174 This herb is often more than three hands-width tall, but its root is like that of a turnip, to which it may fairly be compared, but its leaves are like those of mint. From its branches it sends out a calyx joined to the branch, surrounded with a jacket which it naturally sheds when it begins to turn into fruit. The calyx is as large as the bone of the little finger, but its opening is like a cup. I will describe it further, for those who are unfamiliar with it. 175 Imagine a sphere cut in half, round at the bottom, but with another segment growing up from that base and gradually narrowing and then gracefully widening again at the top, such as in the indented navel of a pomegranate. 176 The plant is surmounted by a top that makes it hemispherical as though carefully turned on a lathe, with notches above it, as I said, like a pomegranate, prickly and ending in a sharp point. 177 The fruit is protected within the coat of the calyx, and is like the seed of the herb Sideritis and sends out a flower which can resemble a poppy. 178 Of this the crown was made, from the back of the head to the temples on either side, but this Ephielis, as the calyx may be called, did not cover the forehead, which was covered with a golden plate, inscribed with the name of God in sacred characters. Such is the adornment of the high priest. (Antiquities of the Jews, Book 3, Chapter 7, §6)
Wearing a "second" crown is how Joshua would have been dressed:
Take silver and gold and make crowns. Place [one] on the head of the of High Priest Joshua son of Jehozadak...The crowns shall remain in the Temple of the LORD as a memorial to Helem, Tobijah, Jedaiah, and Hen son of Zephaniah. (Zechariah 6:11, 15)
During the Second Temple period, did the High Priest continue to wear the crown which was made for Joshua?