Here's my Longines, in perfect shape, in a later WW2 carrying case. These watch boxes looked similar to the Hamilton model 22 wach boxes, but a difference can be seen if compared side to side (If you have both, which I have).
They would have been used on a smaller U.S. Naval vessel such as a Torpedo Boat or a Patrol Boat, with only limited patrol time, where the use of a marine chronometer would have been a waste.
The N and O monogram on the back of the case stands for U.S Naval Observatory which created the specifications for these in 1904 and with various updates they were made and used up through the second world war era.

The Longines museum supplied the following scrap of information:

...please find here below the information I found in our old hand-written registers :

The serial number 2'978'036 identifies a deckwatch in silver.
It is fitted with a Longines manufacture caliber 21.29 that was first produced in 1910.
It was invoiced to Longines-Wittnauer Watch C°, who were for many years our agent for the USA, on the 29.07.1919.
Hoping this information is convenient to you, I remain at your entire disposal for any additional information you may require.

Best regards,

Jennifer Bochud
Museum Curator"