My question is litter for non-covered outside pen? Is this okay to do deep litter method? Are shavings the best?
You can do DLM (deep litter method) in the non-covered pen. What you are trying to do is to replicate a forest floor and those get rained on all the time. You just need to build it a little deeper if it's going to get wet due to the bottom layers being wet, but the top will stay dryish (not soaking wet, but obviously not DRY as it's being rained on).
To do DLM properly, you want a mix of materials that are all a carbon base (wood products, and brown plant life, dead leaves, dried up weeds from your garden etc), sawdust, wood chips, ,mulch, shavings, sticks from pruning trees/bushes, dried grass from mowing (use small amounts at a time, can mold if clumped together in large bunches), dry leaves, hay and straw can be used, but it's better to run it over with a mower to chop it into smaller pieces first, long strands cause matting. Pretty much the best way is to get a variety of materials, at least 3 of the above, then as time goes on, when you add materials, look in the run and see what you can still tell is there, then add whatever is missing. You want different sizes of things and different textures to create air spaces that will keep it all from getting stinky.
After a good rain, when it's all soggy, toss a handful of scratch in the run and let the girls scratch it up and fluff it.