When Oregon joined the Union in 1859, it was the westernmost state—the place where the sun last set on the United States. The Oregon nickname of “Sunset State” has been cited in print since at least 1876.
 
When Washington joined the Union in 1889 and also when Alaska and Hawaii joined in 1959, Oregon lost the “Sunset State” claim as the westernmost state. The “Sunset State” nickname is of historical interest today.
 
   
Netstate.com—Oregon
The Sunset State
Oregon was called “The Sunset State” because of its far west location. At the time Oregon joined the Union, in 1859, it was the most westerly of all the states. This position was lost when Washington joined the Union in 1889 and again when Alaska joined in 1949.(1959 is correct—ed.)
 
Chronicling America
3 November 1876, The New Northwest (Portland, OR), “Letter from San Jose,” pg. 2, col. 5:
I have but just finished reading the account of the triumphant meeting of Oregon Suffragists recently held at Salem, and to say that I exulted in their success and was proud of their energy, is stating a truth very mildly. All honor, say I, to the noble, fearless, and tireless workers in the “sunset state.”
 
ChroniclingAmerica
10 November 1876, The New Northwest (Portland, OR), pg. 3, col. 1:
It was the last Republican gun of the great Centennial Campaign, and its echoes went up and out over the hills and valleys and mountains of the “Sunset State,” inspiring her people with renewed hope for and faith in the great Republic.
 
Chronicling America
4 October 1877, The Daily Astorian (Astoria, OR), pg. 1, col. 2:
The Bee truly remarks: Oregon is the recipient of an honor never before accorded the sunset state.
 
Chronicling America
19 April 1889, The Daily Morning Astorian (Astoria, OR), pg. 3, col. 1:
“The sunset state” is the latest name suggested for our new sister on the north side of the Columbia. That’s good.
 
Google Books
The Significance of Names
By Leopold Wagner
New York, NY: Thomas Whittake
1893
Pg. 35:
Oregon is The Web-Foot State, owing to the humid climate of its coast counties; also The Sunset State, referring to its westerly situation.
 
Google Books
March 1897, Texas School Journal, “State Names” by Thos. D. Dawkins, pg. 106, col. 1:
Oregon: The Web-Foot State. The Sunset State.
 
Google Books
The United States and Their Industries
By William Henry Parr Greswell
London: G. Philip & Son
1899
Pg. 27:
Oregon is called “The Web-foot State,” because of its moist climate, which suits ducks and geese and all web-footed birds. It has also been called “The Sunset State,” because it reaches a more westerly point than others.