Does oil/butter help rise the dough?
Oil or butter are dough enrichments, and they will change the quality of your loaf.  This may be a positive change in flavor and texture, as well as making it stale more slowly, but it is a change.   They will also slow the action of the yeast slightly.

Instead, if the cause of your failure to get a second proofing is that the outside of the loaf is drying out (which from the edited in picture appears to be the case), coat just the outside very lightly with oil or butter, then cover loosely with cling wrap.  I like to spray cling wrap with spray oil, then lay it over the surface of the loaf.

You may also want to improvise a proofing box with your microwave, by boiling some water in the chamber for a few minutes, then using the cavity as the proof box. You still want to oil and cover the loaf while it rises.

This will keep the outside from drying out and losing its elasticity, so that the proofing can continue.

A second possible cause is that the yeast are running out of readily available sugars to digest, and so their action is just slowing down considerably.  You might try adding a teaspoon of sugar to the liquid when you are first making the dough to provide some extra food for the yeast.