Will not change a thing, the hooks are the issue, and need to be wedged/secured.

I have done the tubing "fix" on every H3/H3T seatback we have owned since 2011.
Works every time and never an issue after.

Quick Fix
 

Here's a pic of a simple fix I did 5 yrs ago when I bought my H3. Totally forgot about it until this weekend. Just use a small black zip tie. Haven't had a problem since.
https://cimg9.ibsrv.net/gimg/www.hum...602dfc73b9.jpg

Cool...
 

Like what you did there...
Question. What size hole did you drill in the back plane?

Off hand I don't remember but chose a bit that's just large enough to accommodate the size zip tie you are using.

I used Christmas Tree Clips in the hole to keep the metal retainer in place. Locked in solid, but easy to pop the Christmas Tree Clip out if you want to remove. I actually have the 'back packs', not the 'nets', but same system on both. Also doesn't overstretch the elastic (as with the rubber tubing fix) which eventually will fatigue and you might end up with them loose again.
https://cimg7.ibsrv.net/gimg/www.hum...96351a5735.jpg
Picture without clip.
https://cimg2.ibsrv.net/gimg/www.hum...5f66ec303a.jpg

Picture with clip.
https://cimg8.ibsrv.net/gimg/www.hum...08514cc73d.jpg

The clips on my nets are not flexible and the holes are small. Not sure this would work but it looks like a good fix for your application. I have a huge assortment of thos clips and I'm always trying to find uses for them.

Never a problem over 5yrs later.
8 pan head screws 1/2" long, wedged against the hooks when installing the screws and they don't move at all.
https://cimg2.ibsrv.net/gimg/www.hum...d3217c776e.jpg
https://cimg7.ibsrv.net/gimg/www.hum...a2bf63d12d.jpg

Looks good but I like my zip ties, less obvious.

Both these solutions are great. I copied both tinding the ties worked nice on the upper connections.

However, on the bottom connectors screws are easier then fishing the zip ties thru the holes.

Thanks to both of you for the creative ideas to solve the problem of seat back storage fall outs.