No Jesus did not die "the second death". We all die, then, because of Jesus' redeeming sacrifice, we will all be resurrected and
afterwards be judged whether or not we are worthy of eternal life (Hebrews 9:27, as you quoted). Those that are judged unworthy will not have eternal life, they will die the second death and never again be resurrected - their soul will be killed ("Don’t be afraid of those who kill the body, but are not able to kill the soul. Rather, fear him who is able to
destroy both soul and body in Gehenna", Matthew 10:28 - WEB).
However, Jesus did not sin, and was therefore judged by God to be worthy of eternal life, demonstrated by God resurrecting him to life, exalting him and making him Lord over all living beings (except God, of course), Philippians 2:8-11 (WEB):
(8) And being found in human form, he humbled himself, becoming obedient to the point of death, yes, the death of the cross.
(9) Therefore God also highly exalted him, and gave to him the name which is above every name;
(10) that at the name of Jesus every knee should bow, of those in heaven, those on earth, and those under the earth,
(11) and that every tongue should confess that Jesus Christ is Lord, to the glory of God the Father.
Therefore there is no reason for Jesus to ever suffer the second death, and Jesus confirmed this in Revelation 1:18 (WEB):
(18) ... I was dead, and behold, I am alive forever and ever. ...