If you build the bridge completely before you ever toggle the sign, those blocks will "load" the sign for future use, and you won't need to use a chest at all. Moreover, if you do use a chest, you really only need to use a chest before the first toggle. After that point the blocks get moved to and from the sign. If you look at the sign when the bridge is toggled "off" it will show the number of blocks "stored" in the sign on the top line, iirc.