To achieve such feature I prepared mix which is to be used in both script and cron

The example:

crontab -l
CRON_ERR=/var/log/overseer.log
* * * * * YourScript.sh 2>>$CRON_ERR || echo "[`date`] YourScript.sh" >>$CRON_ERR 2>&1

As I have already mentioned you have to modify your script either

#!/bin/bash
set -E
GLOBAL_ERR=0
trap "GLOBAL_ERR=127" ERR

[... your instructions]

exit $GLOBAL_ERR