There are several William Gardners listed in the records. It is always in your best interest to post the date and place of birth if known with your queries. Or an approximate guess would help also.
I found a William Gardner as an inmate at the on the 1910 Oregon, Marion County Census. He is age 48, single, born Minn, father born Scot/Eng, mother born Canada.
Since your William SR. is in Lane County with his wife on the 1910... and this one's age and place of birth are way off, this isn't him.
Your William J Gardener born 1881 Iowa is in Josephine County: 1910 United States Federal Census > Oregon > Josephine > Grants Pass Ward 2 > District 124
William J. head of house~age 29~married 2 years~Iowa~father born Eng~mother born PA~Jeweler~rented house Florence wife~27~married 2 years~0 children~PA~Eng~CA
It looks like Florence remarried on December 9, 1918 in Josephine County to Edward Cassidy. 1920 United States Federal Census > Oregon > Josephine > Grants Pass > District 211 Cassidy, Edward Head~51~WI~Ire~Ire~mining -, Florence wife~37~PA~Eng~CA -, Agnes G. stepdaughter~9~CA~Iowa~PA
It would appear that the child is William Jr's.
California Birth Index, 1905-1995 Name: Agnes M Gardner Birth Date: 16 Nov 1910 Gender: Female Mother's Maiden Name: Letcher Birth County: Sacramento
Which obviously you must know about since I found this next record: Name: Agnes M. Fuhrmeister SSN: 543-50-3237 Last Residence: 97504Medford, Jackson, Oregon, United States of America Born: 16 Nov 1910 Died: 15 Nov 1993 State (Year) SSN issued: Oregon (1961 )
If you have any other specific questions let me know. Laura