The best bet for newspapers is simply to recycle them after reading them, even the parts you have not read!
Recycled newsprint has many uses, including making eggboxes, cartons for cereal, shoeboxes and a large proportion can be reconverted back into newsprint. In some parts of the world over 60% of the newsprint consists of recycled material.
So by recycling newsprint you may get the chance to read the parts of the paper you missed last time, transmogrified into more interesting news!