'Text file created as log of bash / python not chanigng date

I have a below bash file:

python -u '/home/path_to_dir/run_script.py' >> /home/path_to_dir/log_files/run_script.$(date +'%d_%m_%Y').log.txt 2>&1 &

The bash file is scheduled to run every time the system starts. Currently, the system is on for 2 days continuously. Today I was looking to open the log files but I see there is only 1 log file for date i.e 06_05_2022.

How can we have the log file for each date while continuously running the script? Also, can we have the date & time before the log? Example:-

[07-05-2022 13:55:58] Valid / Error Log



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