I have made the agent the only available agent to receive phone calls ,he then makes a test call from his mobile, to a designated number, but he is taken to a queue and then it connects to another agent who is offline.

Check your Contact Center routing configuration and Omni-Channel Flow logic to ensure that inbound calls originating from an agent's registered phone number or device do not trigger fallback queue routing rules. You should also verify that the telephony provider (such as Amazon Connect) and the associated Salesforce routing flows correctly identify agent-initiated calls and map them to the proper service channel without triggering automated fallback user ownership. 

Then, review your routing configuration in Setup > Omni-Channel > Service Channels and inspect the Omni-Channel Supervisor logs to trace how the call routing attributes evaluate during test calls. Ensuring that your agent's presence status is properly synchronized with the telephony user mapping will prevent the system from defaulting to offline fallback agents and ensure incoming calls connect immediately to the intended available user.