The traditional Japanese origami crane has a fold along the length of each wing, as shown below.
It is a fold that is not necessary because, as you can see from the picture, if it were not there, you’d get a larger, undisrupted surface for a wing. The question then becomes can you fold it without making the fold, without compromising on quality of the rest of the piece. That fold serves as a “guide fold”, as I like to call it, where in one step, you line up an edge to it. If you folded it that way, you’d be eyeballing it, which is not origami at its purest.
Drawing a line on the fold on a finished origami crane, I unfolded it to see where it was on the crease pattern (the pattern of folds to make an origami piece). From that, I refolded it, avoiding folding along the line and succeeded in reassembling it without making that fold. Where that fold served as a guide fold, there were layers underneath which met along that fold that was a centre line at that step (one diagonal on original square), to which I could line up an edge. It was harder as it was a fold inside of the piece at the step, rather than outside of it. That’s the best I can describe it, but forget about that. See the results below! It’s in plain white to show the lack of a fold clearly, but imagine whatever paper pattern or texture you want on the wing, with a fold on the wing like above, or without like below. I think most people would like it without the fold as below.