I have no idea what it’s called now, but according to Neil Brennan, in 1941 it was called “Rice Lane”:
My first thought was that this was taken on the Chemistry side of the loop, but a look at the photo I found in Jim Sims’s papers (and thank heavens I did!) makes clear that it must be the other side:
Aside from the trees there’s not much to go on, but you can see the sidewalk coming from the left and peeking through the trees on the right what looks like one of the dorms. There are a couple of possibilities but I suspect it’s the sidewalk that runs right through what today is the middle of Fondren Library.
Let us pause and just enjoy yet another thoughtful photograph from young Mr. Brennan. Gorgeous cars, too.