If TLS incorporated a feature by which clients can send the server name after the TLS handshake which would cause renegotiation of encryption keys with the requ

If TLS incorporated a feature by which clients can send the server name after the TLS handshake which would cause renegotiation of encryption keys with the requested server (if available) then wouldn't that solve the problem at hand? That way there would be no nees for an unencrypted SNI. Is there already a TLS extension for this? If not,would have been nice if 1.3 incorporated this.