In the last scene where Jack is shaving in the restroom of the sheriff's office, he is wearing the new badge shown in the series, not the one worn by Sheriff Cobb and Deputy Jo Lupo in the pilot. This scene was probably added after the series was picked up for the first year for continuity.
Fun fact: The license plate number of Taggart's jeep has NACL 93 as the license plate number. Na is the scientific symbol for the element sodium, and Cl is the scientific symbol for the element chlorine, which together, NaCl, makes sodium chloride, also known as salt. Also, salt is a part of saline solution which has an osmotic coefficient of about 0.93.
Jack Carter ends up in Eureka purely by chance because he crashes his car at the beginning of the pilot and has to walk to the nearest town (Eureka) for help. Throughout the rest of the series, there is a running joke about how often Carter crashes, damages, totals and even loses his Sheriff's jeep.
Opening placard for the series: "A little knowledge is a dangerous thing. So is a lot." Albert Einstein.
Jack's letter states he is being transferred to Eureka, Washington. It will later be established that Eureka is in Oregon.