The technician should inspect the client's firewall configuration. Since only one specific user is unable to access the server while other users can access it successfully, the issue is most likely isolated to that user's device, account, local configuration, or local security settings. After switch and router logs show no network infrastructure issue, the next logical step is to examine the affected client.
The CompTIA Network+ N10-009 troubleshooting methodology emphasizes identifying the scope of the problem, testing the most likely causes, and isolating whether the issue is client-side, server-side, or network- side. A local firewall could block outbound connections to the server, block a required application, restrict ports, or apply a profile that prevents communication.
Updating network firmware is not an appropriate next step because there is no evidence of a widespread infrastructure problem. Checking the server's DNS configuration is unlikely because other users can access the same server. Checking the server's event logs may be useful later, but the scenario points first to the single affected client. Therefore, inspecting the client's firewall configuration is the best next troubleshooting step.