I had this theory before the current arc, but updated towards it once it became more important to Harry in chapter 89.

In Humanism, Harry thinks about vanquishing future death, but that would not help the majority of the world's population (which has already died). What with the only known method of backwards time travel creating stable time loops, and with people already having died, this makes sense to a degree. But if he were to find a way to upload just prior to death, then technically everybody would have died, but in most ways that count they would not. As people have died more than six hours ago, this also seems impossible with known methods of time travel. However, Harry was (understandably) too distracted when learning about Time-Turners to notice that the six-hour limit could not possibly be fundamental. I believe I have a way around this limit, using only Time-Turners and Muggle understanding of physics. It does rely on the following assumptions:

1. Magic works very far away (many light-hours at least) from Earth
2. Time-Turners deposit you stationary w.r.t. Earth's rotating frame of reference.
3. Magic can reinforce Time-Turners and wizards to be able to withstand incredible accelerations (at least 2000 g)
4. Magic and science combined can maintain a space-worthy vessel at these accelerations
5. Time-Turners take you back the appropriate number of hours in whatever frame you happen to be in when you engage the Time-Turner.
6. Magic and science combined can create space suits that allow survival several light-years from Earth for extended periods of time OR devices that can maintain themselves in such conditions for extended periods of time and can operate Time-Turners.
7. Magic would allow the below scenario to play out (no DO NOT MESS WITH TIME notes). For safety, this should only be attempted after adding explicit protocols for those notes should they prove necessary; that's better than conjuring arbitrary obstacles with Time-Turners.
8. The below calculations are approximations based on my incomplete understanding of special relativity, but could almost certainly be improved by somebody with a better physical understanding than mine.

Not all of these conditions are strictly necessary, but the details of the situation would need to be modified to suit them better. I will now illustrate how to prove to somebody that you can go back in time more than six hours (specifically, six hours and five minutes); extending this to allow for remote unobservable uploading is left as an exercise to the reader.

1. Create a space ship traveling at 88.5% of the speed of light in uniform circular orbit of the Earth, matching the Earth's natural rotation direction and speed. This will require it to be 3.38 light-hours from Earth and accelerating at almost 2000 g, but we're already assuming we can do that.
2. On that ship, have somebody/something capable of receiving messages, using a Time-Turner, and sending messages after using said Time-Turner.
3. Find somebody to agree to this experiment, and to agree on a protocol. Verification can include hashing ("I believe one cannot reverse an MD4 hash, even with magic, in less than 6 hours"), trusted third party ("I doubt that Chief Warlock Dumbledore would lie about what time you gave him a message"), or any other scheme that everybody agrees on.
4. On Earth, get a means to listen to a light-speed message sent by the person/thing in step 2. When it receives a message, tell the person from step 3 and prepare verification, but do not tell the person from step 3 what the message is.
5. Have the person from step 3 wait six hours and one minute before deciding on a message to send, and have them give you this message.
6. Complete verification that the message was received when claimed. The person from step 3 now believes you sent a message back more than 6 hours.
7. Send the person/thing from step 2 the message at light speed.
8. Over three hours later, the person/thing from step 2 receives the message, goes back in time six hours (local reference frame)/13 hours (Earth reference frame), and sends it. Over three hours later, step 4 completes.

EDIT: Fixed the link for acceleration calculation by escaping parentheses