Air Traffic Controller reported inter-facility coordination issue that resulted in a reported NMAC from an aircraft.

Narrative

Fresno Approach coordinated with our Supervisor to protect intersection BRETT for a couple practice approaches into VIS; but never indicated to us which targets/beacons in the vicinity of VIS were the protected traffic. We saw that we had Aircraft Y inbound to VIS so we mentioned that and Fresno Approach said that 6000' at BRETT is fine. I set Aircraft Y up direct BRETT down to 6000' and flashed the handoff to Fresno. Fresno accepted the handoff around the Porterville Airport and I attempted to switch Aircraft Y to Fresno on 118.5. Aircraft Y then said they would rather just cancel IFR; squawk VFR and switch to CTAF to avoid talking to Fresno Approach (a very common practice amongst pilots flying into VIS from the south). I told Fresno that Aircraft Y cancelled and went VFR and they asked if I called traffic. I told them no because all I was doing was protecting BRETT at 6000' in which there was no traffic for at the time of switching. Fresno called us back a few minutes later reporting that Aircraft X on the missed approach out of VIS declared a NMAC with that jet. Upon listening to Fresno Approaches audio of the situation; it seemed that the Aircraft X had checked in with them on the missed approach however the Aircraft X could not receive Fresno Approaches transmissions or control instructions. This led to the two targets getting close to one another but unknown how close.There are a few viable solutions that could have avoided the issue:1. There were a couple limited data block in the vicinity of VIS. I think if I would have had the information on which specific aircraft were doing what specific actions; I could have informed Aircraft Y before switching to CTAF of possible traffic situations ahead. 2. I could have disregarded Aircraft Y want to switch to CTAF and assign him the switch to Fresno Approach for advisories. 3. The radio coverage issues Fresno Approach has for low level traffic near VIS is not safe. Better frequency coverage would likely have avoided this incident.