Looking at the specs on Captain America's shield, its diameter appears to be set in stone at 30 inches, which presented a real problem for Steve Rogers on how to carry the darn thing with him in his civilian identity in order to be ready for action in an emergency. Yet very early during his membership in the Avengers, that problem appeared to be solved, and rather cleverly, at that:
(Good thing airports and offices didn't have metal detectors in 1965--though an Avengers priority card would have smoothed things over for Steve, I suppose.)
There would seem to be no other option for concealing the shield, since there's a trading card out there which confirms the diameter width.
Just for fun, take a metal tape measure and extend it for 30"--snap the lock on the measure so that the tape holds in place at that length, and then hold the tape horizontally against your stomach, making sure the 15" point on the tape rests at the center of your stomach. Then look down and note how far each side of the tape extends on either side of you, and picture that width extension on your back. Conclusion: even Steve Rogers would look absolutely ridiculous walking anywhere with his shield strapped to his back underneath his clothing. (Unless Rob Liefeld had spiked Dr. Erskine's super-soldier serum.)
At some point, though, Steve indeed began wearing his shield on his back in such a fashion, which somehow--somehow--fit without making his back look freakishly out of proportion with the rest of him. Technically, you could say the way had already been paved for that kind of development, since his shield fit just fine on his front side.
Movie-Cap seems to make it work, though we know that Movie-Cap uses several shields at different sizes. But this looks pretty proportional and realistic to me:
Though our stalwart cosplayer here might not get away with it:
It looks like most do-it-yourselfers find it difficult to actually make a 30" shield work, strapped to their back. So the shield is downsized.
At such a size, the shield could likely only be wielded in battle by little kids wearing Captain America Halloween costumes.
Also, let's face it, Jack Kirby should have a fair idea of whether or not something like this could work. And, well, even he seemed to later face up to reality. (Or, in this case, face down.)
I'm still not sure what was wrong with the old artist's portfolio case idea. An attaché case worked pretty well for Tony Stark, didn't it?