How to add a timestamp to bash script log?

You can pipe the script’s output through a loop that prefixes the current date and time:

./script.sh | while IFS= read -r line; do printf '%s %s\n' "$(date)" "$line"; done >>/var/log/logfile

If you’ll be using this a lot, it’s easy to make a bash function to handle the loop:

adddate() {
    while IFS= read -r line; do
        printf '%s %s\n' "$(date)" "$line";
    done
}

./thisscript.sh | adddate >>/var/log/logfile
./thatscript.sh | adddate >>/var/log/logfile
./theotherscript.sh | adddate >>/var/log/logfile