You can't do this with just the aws command, but you can easily pipe it to another command to strip out the portion you don't want. You also need to remove the --human-readable flag to get output easier to work with, and the --summarize flag to remove the summary data at the end.
Try this:
aws s3 ls s3://mybucket --recursive | awk '{print $4}'
Edit: to take spaces in filenames into account:
aws s3 ls s3://mybucket --recursive | awk '{$1=$2=$3=""; print $0}' | sed 's/^[ \t]*//'
down voteaccept
You can even check out the details of AWS CLI with the AWS Certification.