Shaving soap mugs were sold with some type of covering material to keep the soap clean. These covers were initially temporary in nature (paper or flimsy) to protect the product in shipping and while on display in the stores. Later covers were a bit more substantial and could be used by the consumer to cover the mug between shaves. Below is display of these covers according to their material and in a chronological order of their appearance. The listing is from oldest to most recent.

Paper Covers

The original covers were made of a heavy paper or thin cardboard inserted over a crimped paper liner.
paper 1 The first paper soap cover was used on the Hull Pottery mug from 1938 until the early 1940s.
paper 2 This paper cover, shown on  Friendship Pottery mug was used from the early 1940s to 1946.
paper 3 The last paper cover used on the Early Glass 01from 1946 to 1948.

Clear Plastic Covers

clear 1 Clear plastic soap cover used from 1948 to 1955.  Note that there is no longer a paper liner over the soap.
clear 2 This clear plastic cover was used from 1956 to 1962.

Red Plastic Covers

Red plastic covers replaced the clear plastic around 1962 and were used until 1992.  The first two are made of a thin (flimsy) plastic sheet.  The remainder of the examples are made from a thicker, more durable plastic, and appear to be intended for continued use to cover the mug when not shaving.  I have seen these covers only on mugs manufactured for non-U.S. markets (Canada, England, etc.) and I suppose they may have been a requirement in those areas.

thin plastic 2 From 1963 to about 1967 the thin plastic cover showed a printed gold compass rose.
thin plastic 1 From 1967 through the `980s the compass rose was embossed into the plastic
plas 1 This cover has a small tab to facilitate lifting off.  The cover has the words "OLD SPICE" in side a center circle around which is a compass rose.  The lettering and compass rose are embossed as opposed to raised.
plas 2 This cover is like the one above except the lettering and the compass rose are raised above the surface.
plas 3 Graphics are printed in gold on this lid.
plas 4 Identical to the lid above except for the small tab on the edge.