Count times at Bay County Jail typically run at about 10am, 4pm, and 10pm. Each count lasts roughly 20 minutes, during which all inmate movement stops and phone access is suspended until the count clears. If your inmate is not reachable during those windows, that is almost certainly why.
On the question of obtaining phone records for who your inmate has been calling, that information is not available to you. Inmates retain privacy rights over their communications from a civilian standpoint. The facility monitors calls as a matter of institutional policy and that is disclosed to everyone using the system, but that monitoring belongs to the institution and law enforcement, not to family members or outside parties.
There is no public record of an inmate's call history that you can request or access. Private phone records, whether for someone on the outside or someone incarcerated, are not public information regardless of the circumstances.
If the underlying concern is about who your inmate is contacting and whether you can trust how the phone account you are funding is being used, the practical answer is that you cannot verify it through records. What you can control is what you fund. InmateAid's discount phone number is tied to one number only, meaning calls made through that number reach you and nobody else. That does not tell you what is happening on the facility's general phone system, but it does ensure your money is going toward calls to you specifically.