Hi Liz,
I did a search on Ancestry using only "De La" for the last name.
The 1841 census has 14 "De La Cour" names who were born in and are living in Jersey, Channel Islands.
The 1851 census has 10 "De La Cour", 5 indexed as "De La Cowe" but the image looks to me like it's De La Cour, and there is 1 indexed as "De La Croix" which looks right on the image.
The 1861 census which I missed before has 41 indexed as "De La Cour" and 3 "De La Cous".
The 1881 census has 1 "De La Coeur", 70 "De La Cour" and 1 "De La Cours".