Despite his personal admission that he did not visualize details in accouterments well, I believe there was purpose behind all of the professor's statements, and (as with the cloak-and-hood) an awful lot can be inferred of what was going on in his mind's eye by reading what he didn't say.
Here we have a description of Legolas. It doesn't state what sort of quiver it is, or where it hangs persay, but immediately after the quiver is mentioned a long white knife is described...at his belt.J.R.R. Tolkien, in The Lord of the Rings: The Fellowship of the Ring, Chapter 3: The Ring Goes South wrote:Legolas had a bow and a quiver, and at his belt a long white knife.
Here again, this time with men, we have an indirect reference to lean on. Immediately after the reference to quivers, swords are listed specifically with a location: at their sides.J.R.R. Tolkien, in The Lord of the Rings: The Two Towers, Chapter 4: Of Herbs and Stewed Rabbit wrote:Two had great bows, almost of their own height, and great quivers of long green-feathered arrows. All had swords at their sides, and were clad in green and brown of varied hues, as if the better to walk unseen in the glades of Ithilien.
Why describe swords being hung at their sides, and a white knife at Legolas' belt? These seem to be known places for swords and knives...when one thinks of a sword, it hangs from the belt in the mind of the general populous. In fact, it pretty much goes without saying. So why not give a location to the quiver? I believe this is because, in the professor's mind, it followed the same popular familiarity, and went without saying. In short, describing the locations of these blades served as a way to separate the two pieces, placing the quivers in the only logical configuration left: on the back.
This can't be taken as gospel, but there you have it...pretty much the only argument we'll ever have from the text that points in any specific direction. I get where those of you who also do historical work come from regarding the disapproval of back quivers in Western- and Proto-European settings, but in the text here, it is what it is. Well, close, anyway.
In addition, just for grins, I find it hard to imagine a quiver described as "great" on the Rangers in Ithilien being a hip quiver or arrow sock/bag...but that's just me.