The version of the Ten Essentials I've seen already include extra clothes, water, map and compass.

The three things that are hardest to improvise in the field are a knife, fire making and a flashlight, so I'd suggest redundancy in those areas. Redundant navigation (a GPS plus a map and compass) is also highly recommended. I usually drop a waypoint at camp and/or at the trailhead and then turn the GPS off, relying on map and compass from there on. There's a strong argument to be made for having the GPS track your movement, but in that case I'd want to bring spare batteries.

Definitely add cordage to the list. I also like to add a metal cup, which makes boiling water for disinfection or for tea much easier.

I'd add a PLB if possible. They're more rugged and more likely to work than a cellphone.