I guess you are not using the ssh utilities shipped with macOS, the system default ssh-agent daemon should creates a unix socket locates at dir
/private/tmp/ and suffixed with
System default ssh-agent daemon is launched by
launchd and always starts up before Core Tunnel, so the
SSH_AUTH_SOCK var is always ready when Core Tunnel starts.
If your ssh-agent not managed by
launchd, then the
SSH_AUTH_SOCK var may not set correctly for GUI applications.