I don't know if this will help, but I looked on the census lists on ancestry.com for Gustav Olson and found this couple living in Waterbury,New Haven,Connecticut with their daughter Edna and her husband and family. There are a lot of Gustav Olsons in the United States born at the same time your Gustaf was born.

Connecticut is next to New York state.

As far as the picture with the flag and name Spokane...Spokane is a city in Washington State on the west coast of the US. Perhaps they were vacationing there?

Name: Gustaf Olson
Home in 1930: Waterbury, New Haven, Connecticut
View Map Age: 63
Estimated birth year: abt 1867
Birthplace: Sweden
Relation to Head of House: Father-in-law
Spouse's name: Anna
Race: White

Name
Edward Gros 31
Edna Gros 29
Edward Gros 8
Ruth Gros 7
Dorothy Gros 4 10/12
William Gros 2 10/12
Robert Gros 1 7/12
David Gros 7/12
Gustaf Olson 63
Anna Olson 62

Hi Morgan,
I just sent you JOHN N. TILLY's petition for naturalization via ancestry.com, so please make sure you look in your spam folder so that you don't delete it. I have been away on personal business, so I am sorry it has taken me longer than I expected in getting back to you. I will send you whatever I can find today, so look in your personal email box, and not on here. Kathleen~CT.

Only one daughter listed on the 1930 census.

FROM THE 1930 CENSUS - LONG BEACH, NASSAU COUNTY, NEW YORK, USA
Albert A. Lannin, head - 31
Ethel Lannin, wife - 34
Dorothy Lannin, daughter - 9
Annie Lannin, mother - 50