What is it called where you start with a single letter that is a word ('I'), and then you add a letter, and possibly rearrange, to make another word ('IT'), and then you keep going ('TIP', 'PITH', 'PITHY')?

I was scanning some job ads, and kept having my eyes drawn to the word 'DIRECTOR'. I realized that it has all the letters of 'EDITOR' in it. So I spent a few minutes coming up with this, although I mostly worked backwards.

I
ID
DIE
TIED
TIRED
EDITOR
CORDITE
DIRECTOR
DIRECTORY

And I admit that I used the Internet Anagram Server to find 'CORDITE'.