Maybe I'm not understanding the question. Do you mean use a bag to transport a box and a stamp? Why not put the stamp inside? If it won't fit, it's not the "perfect" box IMHO.

could you just throw a bag around the box with the HH in the bag?

Sometimes a plastic bag on the outside will trap moisture and cause condensation in the box, damaging the original log book. It's generally not good to add anything to a box that isn't already there (excluding the HH of course), but only to repair as needed.

It's best to use a smaller container that will fit, or find a larger traditional LB, as already suggested.

Artemis

Most of the HHs I have sent out have been in one freezer ziploc bag. Usually a small sized one. They seem to survive and can then be put in a pouch sometimes. A pill bottle wouldn't fit in many we find in our area.

Also try and put it in a well trafficed box so it gets moving quickly.

Enjoy.