In one aspect, a shade assembly may generally include a shade fabric having a front side and a back side. The shade assembly may also include two or more batten spacers, with each batten spacer being coupled to the shade fabric at spaced apart locations along a vertical length of the fabric. Moreover, the batten spacers may be substantially aligned with one another horizontally along the back side of the shade fabric. When the shade fabric is moved to an opened position, the batten spacers may be configured to vertically engage one another to form a vertical spacer stack along the back side of the shade fabric.