You Do Need power plug adapter while traveling from New Caledonia to Saint Barthélemy.

Here is Why You Need An Adapter

The outlet and plug type used in Saint Barthélemy are C and E 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

Saint Barthélemy

In Saint Barthélemy, outlets of type C and E are in use.

Type C

Type E