How to rename a custom taxonomy slug?

I’ve got an existing custom taxonomy where the URLs look like:

domain.com/foo/[tax_term]/

…and I’d like to change it so that the terms can instead be found at:

domain.com/bar/[tax_term]/

I think the ‘renaming’ bit can be done by simply changing the ‘rewrite’ parameter where I register the taxonomy, from 'rewrite' => array( 'slug' => 'foo' ) to 'rewrite' => array( 'slug' => 'bar' )

…but I’m wondering if I’ll need some sort of redirect afterwards to stop existing links from other sites and search engines breaking.

Does anyone know how best to achieve this?