Many people, reviews, websites, etc.---even the DVD box refer to Gillespie as the Sheriff. He is however, not the Sheriff. There are Sheriff's men (maybe even the Sheriff himself) depicted in the capture of Harvey Oberst scenes (wearing khakis and sporting "County Sheriff" shoulder patches). However, Gillespie is Chief of Police for the City of Sparta.
shareI've never noticed this but you are absolutely correct. Gillespie would have had jurisdiction over the city of Sparta but limited power (if any) outside the city limits.
I guess we have to assume that the factory site was within the city or had been annexed, otherwise technically the Sheriff would have had jurisdiction after they discovered he had been murdered there.
I wonder if some of the confusion comes from the TV version starring Carroll O'Connor. In the later seasons of that show Gillespie goes from being the Chief of Police to Sheriff. This was when O'Connor eased back on his role, I think for health reasons.
Normally police officers have peace officer authority anywhere in the state. Since the murder victim was found in the city limits Gillespie and his department would have had jurisdiction.
The location of the actual murder is not determined until later in the investigation. My guess is the elected sheriff would have been very happy for Gillespie to continue to work this "hot potato" of a case.