When Carter and Fusco enter the diner at the beginning of the episode, the camera from behind them shows Mr Reese sitting at a table without any drinks on it. In the following frontal shot of them sitting down, suddenly two drinks appear out of nowhere.
The Texas flag outside of the bar "The Razorback" is hung upside-down! The red color of the flag should be on the bottom, it represents bravery and is symbolic of the blood that was shed.
Barbara Russell states that Trent Russell was murdered in the 1990s. When the murder is shown in a flashback, the SUV driven by the murderers has a Texas license plate base first issued in 2009.
There is no way the bank would tell Mr Reese where someone used their credit card unless he was a police with a specific type of warrant.
Mr. Reese had no way of telling where Root's house is. All he knew was what town she was located in and no address information yet he was able to find it.
As soon as Mr. Reese hears about the story of Hanna Frey's disappearance, he automatically concludes that this has to be related to Root without any reason or evidence and decides to go to Texas to investigate it to find Root. It would have been impossible for him to reach to that conclusion with the knowledge he had at the time. He had no reason to connect the two cases which appeared to be completely unrelated. There is no explanation as to how he reached that conclusion.
The entire storyline around investigating what happened to Hannah in Texas is a major plot hole. Mr. Reese already knew Hannah's case was connected to Root, otherwise he wouldn't have gone to Texas to investigate the case. At the same time, it would have made no sense for him to be surprised by finding out Root's connection to the case. He went to Texas in the first place because he already knew. Why would he go to Texas if he didn't already know the two cases were directly connected.