With all due respect to the Consolacion Police Station, a suspect for a homicide incident on January 1, 2020 at around 7:30 P.M. cannot be the subject of a hot pursuit operation more than 48 hours later under circumstances that would indicate that the police lost track of the whereabouts of the suspect (hence the appeal to the public).
In law enforcement, “hot pursuit” refers to an immediate pursuit by the police such as a car chase (see Lumanog v. People, G.R. No. 182555, September 7, 2010). Following this logic, a pursuit where the police never lost track of the suspect even though they cannot arrest him yet for logistical reasons (e.g. the difficulty of getting to for example a mountain hideout – but it is known that the suspect is in the hideout) may be justified as a hot pursuit even if it stretches a bit the “immediacy” aspect of hot pursuit. (A police official divulged to me that he justifies a hot pursuit by coming up with update entries in the police blotter, to document what was being done to track and arrest the suspect.)
But in this specific case, it seems to me that the police had already lost track of the suspect (otherwise, why would they appeal to the public for help?). The pursuit is already broken; there is no pursuit to speak of anymore. The less so then that there is still a valid hot pursuit.
The proper case should just be filed through regular preliminary investigation.