1 Samuel 16:19 Therefore Saul sent messengers to Jesse and said, “Send me David your son, who is with the sheep.” 20And Jesse took a donkey laden with bread and a skin of wine and a young goat and sent them by David his son to Saul. 21And David came to Saul and entered his service. And Saul loved him greatly, and he became his armor-bearer. 22And Saul sent to Jesse, saying, “Let David remain in my service, for he has found favor in my sight.”
Saul had interacted with Jesse twice and was impressed with David.
1 Samuel 17:58 "Whose son are you, young man?" Saul asked him. David said, "I am the son of your servant Jesse of Bethlehem."
After David killed Goliath, Saul wanted to know David's father' name. Why did he ask at that particular time?