Another way of looking at the OP question is to ask oneself, is there a passage in the Bible that expressly states, Jesus died the second death?
No.
" Then I saw a great white throne and him who was seated on it. From his presence earth and sky fled away, and no place was found for them. And I saw the dead, great and small, standing before the throne, and books were opened. Then another book was opened, which is the book of life. And the dead were judged by what was written in the books, according to what they had done. And the sea gave up the dead who were in it, Death and Hades gave up the dead who were in them, and they were judged, each one of them, according to what they had done. Then Death and Hades were thrown into the lake of fire. This is the second death, the lake of fire. And if anyone's name was not found written in the book of life, he was thrown into the lake of fire. " The Book of Revelation chapter 20 verses 11 through 15