1920 Emma and William Woolsey are in Appanoose County, Iowa
This looks like "your" Emma in 1940:
1940 Madison, Clarke County, Iowa William H. Smith, 63, born about 1877 in England, not a US citizen but has first papers Emma Smith 53, wife, born about 1887 in Kentucky Residence in 1935: Same House (for both)
I didn't see any other candidates for Emma King Woolsey.
If this is the correct Emma, then she and William Smith had no children (or none who survived); the children would be under age 10 in 1940.
I searched Find a Grave and Iowa Gravestones Project for William Smith's burial place but did not find him. I did not find him in the same cemetery as Emma (there's a Find a Grave memorial for her, as you know). However, these online databases are not complete. You could write to that cemetery and ask if a William H. Smith is buried there.