Here's an outdoor cage suggestion

[vana]

I've been using a Ferret Nation cage as my skink's summer home for probably about nine years or so. I keep it on my back porch. It is a very well made product, and the price is quite reasonable for how nice it is. I have the single unit, but you can stack multiple ones on top of each other for a skink apartment building. It is great for sunning, and he seems to like it... He has not had a problem rubbing his nose (your results may vary).

You will need to have a shaded area for it, out of the rain; I have a roof over my porch.

Two possible drawbacks:

1) You will need to bring the skink in when it is too hot or cold (obviously)
2) There is the possibility of wild animals bothering the skink, like a snake, or a cat reaching through the bars. If you have a screened in porch, that would be an ideal place to put it.

This is the unit I have:

I see that they actually have a model (Crittern Nation) with less space between the bars now:

That might be safer to keep wildlife out (and for smaller skinks) but it might encourage nose rubbing, I don't know.

Here are some for sale (this is not a suggestion on where to buy one, I don't know the best price, this is just so you can see the options):

https://www.ferret.com/cages/cages/887/