Jessup went'n turned snitch, and that's only the biggest ancient no-no of all, ain't it?" Teardrop assumes Jessup is hiding for his life and whether it be from the sheriff or from Thump Milton's gang, Ree's meddling could potentially expose him and get him killed. Although seemingly harsh, Teardrop's intention was simply to scare Ree into keeping her mouth shut for her and her father's sake.
Teardrop later goes on to become a heroic figure in the plot. Even though Ree blatantly disobeys him and goes to Hawkfall, Teardrop is there to rescue her after she gets beaten half to death. He puts himself in harms way for Ree and at the end of the plot it becomes clear that he will end up getting killed for helping her. He told Ree, "I'll help you some, take your back so you can find his bones, but the deal is, even if you find out, you can't ever let me know who did the actual killin' of my brother. Knowin' that'd just mean I'll be toes-up myself purty soon, too." Teardrop knows if he ever found out who killed Jessup, he wouldn't be able to stop himself from going after them. Even though Jessup made a mistake that earned him grave consequences, he was still Teardrop's only little brother. In the end, Teardrop tells Ree, "I know who now.", and drives off to his assumed death.
After Ree's curiosity had begun to stir up a bad reputation for the Milton family, a distant relative to Ree, Blond Milton, paid her a visit. Blond drove Ree out to an old, burned-down house and he claimed that Jessup died in there during a faulty meth lab. Ree knew that he was lying. Between the "chin-high" horseweed surrounding the house and the fact that her father was the best meth cook in town and had never messed up a lab before, Blond could not fool her. Blonde Milton was Jessup's first cousin, but he was also related to Thump Milton, the leader of the gang that is presumed to have something to do with Jessup's disappearance.
