How to export multiple CloudWatch logs into on text file in an S3 bucket

I'm developing a Lambda function that is suppose to send the messages of multiple CloudWatch logs into one single text file stored in S3.
With the use of Python, here's the code I've compiled:

import boto3, json

client = boto3.client('logs')
s3client = boto3.client('s3')

def lambda_handler(event, context):
	stream_response = client.describe_log_streams(
		logGroupName = '/LogGroup',
		orderBy = 'LastEventTime'
	)
	
	for log_stream in stream_response['logStreams']:
		latestlogStreamName = log_stream['logStreamName']
		
		print(latestlogStreamName)
		
		response = client.get_log_events(
			logGroupName = '/LogGroup',
			logStreamName = latestlogStreamName
		)
		
		print(json.dumps(response, indent = 4))
		
		for log_events in response['events']:
			log_messages = log_events['message']
			#print(log_messages)
			
			s3client.put_object(
				Body = log_messages,
				Bucket = 'S3_Bucket',
				Key = 'Bucket_Object/TextFile.txt'
			)

It does work fine, but when I go to S3 and check up that file in the bucket, I only see the latest message displayed. Then, I noticed that the file has older versions, each contain a separate message.

Is there a way I can export all the messages into one single text file in S3?
How should I export the messages properly?

s3client.put_object overwrites the file. S3 doesn't allow to append to files, you need to collect all the logs that you query from CloudWatch and create a separate, new file for each.

I would also suggest to disable versions, except you have a business/legal requirement to do so.