“Is the hand of Joab with you.” That David immediately suspects Joab in this plot shows us that there is significant backstory that is not in the Bible. David and Joab were close, and Joab certainly must have on occasion shown his desire for the royal family to be united. That would explain why David only asked about Joab and no one else; after all, Absalom had a number of friends and followers who must have wanted Absalom back, why suspect just Joab? He must have let David know he wanted Absalom back.
On her part, the woman would not have had much knowledge of the backstory. She had to be a woman David did not know or he would have known her tale of two sons was a lie. This explains why when David immediately suspected Joab, she said David had wisdom like an angel of God (2 Sam. 14:20), basically saying that because of his close relationship with God, David had access to secret knowledge such as the angels have.