Once you have carried out the selection, you can export the selected feature to a new layer.
How: carry out your selection process > in the contents pane [or, on ArcMap: the table of contents], right click on that layer's name > Dara > Export [features]
You can also continue as you have done previously i.e. create the temporary layer > then export that layer as a new, independent, permanent layer.
How: similar to the previous method, simply right click the new layer > Data etc.
You could put together a quick model / tool that carries out the desired selection and creates the new permanent layer, all in one!
Excelsior.