Skylane 182/RG Turbo Skylane/RG
Air Traffic Control Fully Certified
Situational Awareness
ATC Issue All Types
Air Traffic Control Issued New Clearance
Human Factors; Procedure
A Cessna made an initial call and after the second time I got the called right. The Cessna indicated he was 29 miles north of ZZZ; diverting to ZZZ due to possible fuel leak. I query at what altitude he was at. The Cessna responded with 10;300 MSL. At which I looked and assumed it was the VFR target around that area indicating 10;600 MSL; which is approximately 35 west of my airspace. I assumed at that point it was the Cessna and suggested he contact Center; the sector he was in. The Cessna took the frequency change. I believed I was too quick to transfer communication of the Cessna to Center as he was in their airspace according to the position he provided and what it look like a RADAR target with similar altitude and position. I believed I made an error in judgment. What I should have done was give the Cessna a squawk code; identified the airspace and ask if the Cessna needed additional assistance other than just basic VFR flight following. There after; I could have called Center to see if you would like to handle the Cessna or just leave to me to handle as he was near his diverted destination of his flight.
ARTCC Controller self critiqued the handling of an aircraft with fuel issues requesting VFR following; the reported acknowledging the aircraft should have been identified before issuing changing the aircraft to an adjacent Center.