There is another (?) possibility. The CA DEATH Index is online and it contains a Kenneth Thompson, born in Kansas (1913), died in Los Angeles (1944). His mother's maiden name was POOLE. Father's last name was Thompson.
You can look at the details on rootsweb. I couldn't find the same record on Ancestry (funny!).
Hope this helps.
Do you have the obit for Cecelia Garrett (death: 1965, KS)?
There might have been other children that are still living.
What about her second husband?
There are two possibilities in the SSDI - both born in the 1890's. One died in Wichita, the other in Burlington.
Of course, my first bet is on Cecelia (if this is the right one, of course).
There is/was an elder Earl in Cherokee. Also one in Topeka.