yes - put cover on the back, and the sides if you can
climbing the glass is pretty normal for some of them to do. they do not understand what glass is, so it feels like open space to them. bts are often very stubborn, so once theyre set on going from point A to point B they wont stop till they have to breathe or to think about why it wont work to go through the glass.
when stressed or when looking for more treats or any the like, some bts may climb to see if it works, and some just wont give up. adding something that gives the feeling of extra walls, gives extra security. but you may still find your skink climbing on the non covered side for a while yet. just keep an eye on him/her so he wont fall badly (they usually handle it very well).
since you seem to have a very active skink, i bet he/she is just too curious to think about food xD
