When I make stock, I use a potato masher to try and extract some of the juice from the remaining softened vegetables after siving. Can I use a food mill to do this more effectively?
I don't see why not.
A food mill is regularly used to make purees. It's a popular choice for people making baby food. Here's an example with two different mesh options:
Another option might be a ricer, which is more like a giant garlic press, whereas a food mill has a crank, a ricer has a lever-action.
Ricers are also popular for baby food and mashed potatoes and often have different hole size options.
As a note, you'll want to make sure that there's not anything too hard in the stock like bone fragments. They might be fine in the ricer but they may be problematic in the food mill.
Based on comments here, it seems like the mill is more versatile. Everyone loves the ricer for potatoes but seem to prefer the mill otherwise, so your idea may be better.
User bobcatsteph3 says:
I only use my ricer for potatoes. With the food mill you can puree soups, make smooth tomato sauce, make easy, fantastic applesauce, and yes, mashed potatoes. And you can do all those things with the skin/peels on.
The food mill is harder to clean, and takes more storage space, but I really like OXO's, it's very easy to use, if I could only have one, I'd take the food mill.