How to fix mitts so your toddler can't get them off

This fall my cute little toddler did not like to keeping her mitts or boots on when we were outside. We walk Lilo to school everyday and I would fight with her to keep them on. It would be below zero (Celsius) outside and Isla would have bare hands and feet and I would look like the horrible mother.
Luckily as it got colder and she was wearing snow pants she would at least keep her boots on but not the mitts.

Finally before Christmas I realized that I need to figure something out now since we were visiting family up north where they had lots of snow.
Here is the 2nd attempt at the mitts.I found an old pair of socks, they were very worn at the foot but the rest was in good shape. I cut the foot off.

Then with right sides together sewed the top of sock to the mitt.

And there you go. The elastic on the sock keeps them up and now I can easily slip a coat over. This also prevents her getting snow up her sleeve. I know there are longer mitts out there but the elastic in the sock makes it harder (not impossible) for her to remove them. Also you can put the mitts on over the coat if they do come off. They can easily stretch over top.

As I said this was attempt #2 - the first was with the bottom of a sleeve of a sweater but there wasn't any spandex in it so it stretched out of shape and was slipping off. I took it off and tried this sock instead, I'm thinking it should work better.