First lets put 1 Corinthians 15:24-28 in context.
1 Corinthians 15:20-28
20 But now Christ has been raised from the dead, the first fruits of those who are asleep. 21 For since by a man came death, by a man also came the resurrection of the dead. 22 For as in Adam all die, so also in Christ all shall be made alive. 23 But each in his own order: Christ the first fruits, after that those who are Christ's at His coming, 24 then comes the end, when He delivers up the kingdom to the God and Father, when He has abolished all rule and all authority and power. 25 For He must reign until He has put all His enemies under His feet. 26 The last enemy that will be abolished is death. 27 For He has put all things in subjection under His feet. But when He says, "All things are put in subjection," it is evident that He is excepted who put all things in subjection to Him. 28 And when all things are subjected to Him, then the Son Himself also will be subjected to the One who subjected all things to Him, that God may be all in all. NAS
In Acts 5 Peter tells us that God exalted Jesus to His right hand as Prince and Savior. In Revelation we read the Son is the King of Kings, in 2 Timothy 2:12 and Revelation 20:6 we read that those who overcome will reign with the Lord. At the end of the Lord's millennial reign when Hades is caste into the lake of fire there will be but one kingdom, the Kingdom of Heaven where the ultimate authority is God Himself. Therefore as a dutiful Son the Lord will turn over His kingdom to His Father. This in no way indicates the Lord will not continue to reign as King.
The word "king" in Scripture indicates a person of authority. Herod was King of Judea under the Roman Emperor. Darius was King of Persia who was subject to his father Artaxerxes who was kings of kings over Persia (Ezra 7:12).