There is an example of using reindex to modify the document in the docs here. It changes the name of a field but you can use the same script construct to do what you want to do. I suspect the script looks something like "script": "ctx._source['timestamp_hour'] = ctx._source['@timestamp'].getHourOfDay(). You are right that it will be much better at search time to use this new field. Watch the time zone, btw. I believe the hour that you get in this case is UTC.
You could also do this with _update_by_query because it just introduces a new field which is a change you can do on an existing index. You'll end up with deleted documents in your index but merge should remove them in time. Not all of them, but a bunch. That might be easier to deal with depending on what you are doing.