You Do Need power plug adapter while traveling from New Caledonia to Cayman Islands.

Here is Why You Need An Adapter

The outlet and plug type used in Cayman Islands are A and B and in New Caledonia are C and F. This is how they look like.

New Caledonia

In New Caledonia, outlets of type C and F are in use.

Type C

Type F

Cayman Islands

In Cayman Islands, outlets of type A and B are in use.

Type A

Type B