The 'not of works' is the works of the law which are the purification works, for which circumcision is the sign.
Just read how Paul starts talking about circumcision in the flesh.
Ephesians 2:11
Wherefore remember, that ye being in time past Gentiles in the flesh, who are called
Uncircumcision by that which is called the
Circumcision in the flesh made by hands;
We have to come to him and confess our sins and repent of them.
We have to do what he says.
Salvation had not happened when the jailer believed; for if it had, he would not have asked that question he asked, he would have been saved immediately, according to your mistaken beliefs.
Those people had to hear the message. The message is about repenting of your sins to have life.
See
Acts 10:44,
Acts 11:15,
Acts 16: 32 Then they spoke the word of the Lord to him and to all the others in his house. 33 At that hour of the night the jailer took them and washed their wounds; then immediately he and all his household were baptized.
Did you hear that?
They had to hear the message that saves and it is the message that says to repent of your sins.
His words tells us WHAT TO DO TO GET SAVED and what to do to STAY SAVED.
John 8:51
I tell you the solemn truth, if anyone obeys my teaching, he will never see death."
If you say/confess with your mouth that Jesus is your Lord, then you had better be obeying him.
Luke 16:46 "Why do you call me, 'Lord, Lord,' and do not do what I say?
Matthew 7:21 Not everyone who says to Me, 'Lord, Lord,' will enter the kingdom of heaven, but only he who does the will of My Father in heaven.
Read what Paul says about if you say Jesus is your Lord:
2 Timothy 2:19 Nevertheless, God’s solid foundation stands firm, sealed with this inscription: “the Lord knows those who are his,” and, “Everyone who confesses the name of the Lord must turn away from wickedness.”
The foundation is repentance of sins.
Believing is a works we have to do, and it is made complete by an action, like confessing your sins with your mouth.