IMHO you should keep your attribute set name to be called 'default' as there are circumstances when (if 'default' is not found) another attribute set called 'default' will be created.
However, you are where you are. On Magento Connect there is an extension to update the attribute set of a product (i.e. to change it).
Other attribute set gotchas include configurable products where the daughter products do not show up because they have a different attribute set name to the parent product.
(I believe this works on 1.9.x and EE version 1.14.xx)
Usual caveats apply - test on your development environment first (or take a dump of your database). I have not had problems with this extension, it has 'got me out of jail' a few times over the years.
If you do not like to see the attribute set listed in your products grid, clone the Grid.php file into the local codebase and edit out the column.