I reinstalled ESET Management Agent on the new ESMC server. Here is a clip from the trace.log showing that the Agent service on the new ESMC server is trying to reach the old ESMC server!
2020-02-26 20:03:16 Information: CSystemConnectorModule [Thread 2c14]: StatusLog_NETWORK_IPWINSSERVERS_STATUS: "Rows":[]
2020-02-26 20:03:16 Information: CSystemConnectorModule [Thread 2c14]: Retrieving installed applications list
2020-02-26 20:03:16 Information: CAgentSecurityModule [Thread 103c]: Agent peer certificate with subject 'CN=Agent at *, OU=Operations Support, O=DawnSignPress, L=San Diego, S=C