I first noticed this phrasing in Sam's famous speech in the The Two Towers movie.
And when the sun shines it will shine out the clearer.
The highlighted part seems to originate in the second book of The Return of the King, when Sam and Frodo wake up in Ithilien
‘A great Shadow has departed,’ said Gandalf, and then he laughed, and the sound was like music, or like water in a parched land; and as he listened the thought came to Sam that he had not heard laughter, the pure sound of merriment, for days upon days without count. It fell upon his ears like the echo of all the joys he had ever known. But he himself burst into tears. Then, as a sweet rain will pass down a wind of spring and the sun will shine out the clearer, his tears ceased, and his laughter welled up, and laughing he sprang from his bed.
The Field of Cormallen, ROTK, J.R.R Tolkien
But here are some other examples:
The world is indeed full of peril, and in it there are many dark places; but still there is much that is fair, and though in all lands love is now mingled with grief, it grows perhaps the greater.
Lothlórien, FOTR, J.R.R Tolkien
I beg you to remain and ride with my brother; for then all our hearts will be gladdened, and our hope be the brighter.
The Passing of the Grey Company, ROTK, J.R.R Tolkien
Aragorn stood beneath his banner, silent and stern, as one lost in thought of things long past or far away; but his eyes gleamed like stars that shine the brighter as the night deepens.
The Field of Cormallen, ROTK, J.R.R Tolkien
To me, it seems like an odd turn of phrase, but I'm thinking it must be correct since Tolkien was a scholar of languages. I'm curious, does anyone know examples of this kind of phrasing outside the Lord of the Rings? Is there a name for it?