In chapter 4 of The Scarlet Letter Chillingworth poses as a physician to enter Hester's cell after having witnessed the woman's shameful scene at the scaffold, which led the pair to recognize each other.
When he entered the cell he actually aided Hester, and he even admitted that it is no surprise that Hester had sinned the way in which she did considering how Chillingworth insisted on marrying her, with young age and beauty, while he was "misshapen", old, and devoted to philosophy.
Yet, he does ask Hester the name of the man who "wronged [them] both", perhaps hoping that his help and his kind words would have moved Hester to tell the truth. When Hester flat-out refuses to tell Chillingworth the name of the man who fathered Pearl, Chillingworth responds that he will find this man, because he has a sensitivity towards people that will let the guilty man show himself even if it is involuntarily.
I come to the inquest with other senses than they possess. I shall seek this man, as I have sought truth in books... There is a sympathy that will make me conscious of him. I shall see him tremble. I shall feel myself shudder, suddenly and unawares. Sooner or later, he must needs be mine!
Therefore, Chillingworth's method was to use his inquiry and analytical skills as a scholar and as an extremely bright man to make sense of the attitudes and behaviors of others until he reaches the actual suspect. Chillingworth is a man of extraordinary intellectual gifts and he is quite aware of them. He already knows that he is quite superior in intelligence than all the villagers put together. Hence, he knew that it will take very little for him to catch the indicators that would give away the secret that Hester tries to hide.