Can a Git hook automatically add files to the commit?

I'd like to add an automatically generated file to the same commit using a pre- or post-commit hook in Git, dependent on the files that were modified in that commit. How would I go about this?

I've tried this as a pre-commit hook, but no luck:

#!/bin/sh
files=`git diff --cached --name-status`
re="<files of importance>"
if [[ $files =~ $re ]]
then
  echo "Creating files"
  exec bundle exec create_my_files
  exec git add my_files
  exec git commit --amend -C HEAD
fi

This successfully adds them to the repository, but does not add them to the commit. I've also tried using the last two exec lines in a post-commit hook along with the pre-commit inspection, but no good either.