Like Mattew and Susan, I also found sources for your grandfather Willoughby.
I went ahead and took the step of adding everything to his profile. There are enough links to sources that you can add his parents and siblings.
From that information, you can use FamilySearch to go back further. Sometimes you have to get creative in how you search. For example, I could find the 1940 Census and 1930 Census for Cody Willoughby, but nothing further back showed up in the search results. But using his father's name, and rough birth year, then I could find the 1920 Census, and then find Cody within that result.
Another tip is when you search on someone's birth year, don't search on an exact range. Do +/- 1 or 2 years. For example, Cody was known to be born in 1914. When doing a search, search for birth year 1913-1915. Or 1912-1916. Census records only give an *approximate* birth year. Which could be off by a year. When you widen the scope, this allows the search to find more records that might be useful, then if it was a narrow range.
The other big thing that helps in searching is having some idea of location.
For example, I found the 1930 Census for your grandmother Marie (Cody's wife). But I found what looks to be the right match because this particular 1930 Census record was for Precinct 1, Van Zandt, Texas. This location showed up in other sources for Cody, hence this is probably a match. This is coupled with the right name, and approximate birth year for her mother.
Now that you have the 1930 Census for Marie, you now have her missing father: His name, approximate birth year, and the state that he was born in. Sometimes this can be enough to go on to start researching where he came from.
Going to your mother's side is more of a challenge. The profile of your maternal grandfather, Clarence Smith, does not have any location at all. And of course only, the decade he was born in. This is ok, because of the privacy rules. However, going to his father, it's going to be really tough: A first and last name, both of them fairly common, with only birth year and death year. Without more information, like a location of some sort, for birth, death, marriage, residence, then it will be challenging to find the right match. Because then the only other information we have to match on is his wife name.
Any more information that you can get and add would be helpful in tracing it back further.