Is there anything I can add to a standard no-knead bread recipe to allow the bread to stay softer for longer? Bread baked the night before, while still edible, is not as pliable the next midday.
I currently use bread flour, yeast, salt and water but was thinking of adding milk, butter or eggs. If there is a single food additive I can add I would consider this, but I don't want to adulterate the recipe too much.
Ken Forkish's Flour, Water, Salt, Yeast has a brief note on storing baked bread (page 77).
I got over my aversion to storing bread in plastic bags many years ago, after trying all the alternatives and realizing nothing else keeps the bread as well. The crust will soften, but the bread won't dry out. The straight dough breads will keep for two or three days. The breads made with pre-ferments will keep a day longer than that, and the levain [meaning sourdough] breads from this book will keep for five to six days, if you don't eat it all before then!
I propose sourdough starter as an additive that should almost double the life of your bread. Unlike the fatty enriching agents you suggest, using a sourdough starter should minimally impact the type of loaf you are baking. The process for making sourdough bread is quite similar to the "no-knead" method you are familiar with, and the result is similar, albeit with a more complex flavor. The downside, of course, is the work involved in maintaining a starter.
With respect to using a plastic bag, I usually split the difference: I store the bread uncovered for the first day to enjoy the crunch of the crust, and move it to plastic after to keep it moist. Do make sure the bread is completely cool before putting it in plastic.
I would also encourage you to embrace stale bread as a wonderful ingredient. When I bake, I count on a portion of the bread going stale, and use it for toast, croutons, french toast, stuffing, panzanella, or bread pudding.
Adding any or all of your suggestions -- milk, butter, and/or eggs -- will likely soften the resulting bread. They will also cause it to remain softer for a longer period of time.
But they will also tend to change the flavor and texture of your bread. If you're used to baking a crusty bread with only flour, water, yeast, and salt, the bread with enriching agents like milk, butter, and/or eggs won't have the same crust and it may not rise the same way (as these ingredients can weigh the bread down). Furthermore, you may need to modify baking conditions if you bake at a high temperature, or else risk excessive browning or burning of the exterior.
Instead of any drastic modifications, if you really just want to keep the bread a little softer, you might consider your storage method. Putting a lean crusty bread into a plastic bag (or other sealed container) will ruin the crustiness but also will keep the bread much softer for several days.
