bkay

The only sensible option (my opinion) is to replace all the veneer or leave it alone. If you peel it off, the wood under it is not pretty, nor can it be made pretty. It's a substrate. It's the "bones" of the cabinet.
Modern Veneer is 1/2 the thickness of what was available in 1940, so blending in half of it won't work. Some folks use two layers if they are patching. Replacing a big piece like that takes skill and patience.
No one is looking at the back, anyway.
JMHO
bkay

Maybe you could inject glue under the veneer and reglue it. It would probably take steam to get it to lay down --- it's a thought. Look on youtube. If you have a local Rocklers, ask them.

JoeJr

I invited my craftsman neighbor over yesterday evening to take a look. He gave me some suggestions on getting it glued down better. I thought about steam to soften the wood, I guess via the steam iron. Otherwise I will leave it alone, because the back will be against a wall most likely. And it's not super horrible, I'll post a picture later on.

JoeJr

Here's the back. I think that larger gap can be glued down pretty easily.

AlZilla

Yep, mine has a case of the peelies, too. I'm thinking of syringing some glue in there and weighting it down. Normally when you apply wood glue, you sand the surfaces receiving the glue. I just don't know how to accomplish that. I guess I should head over to the woodworkers forums and poke around.

1.41

Beautiful.