It's the end, but the code-golf has been prepared for
Python 2, 340 bytes

#coding:L1
import zlib;r=zlib.decompress("x5mÂ@¯Â 8]c]u×Xã-q1H5½ýRãþ#÷O³PÌsÆ=¹IºÁÑêx¼*~W=¿ñX,èÆ{v6ø¢§&lt;â¦©Yô³é¦üäÇ¼Ý¥¦NAs+Ò5(´£96ÉÃë}»eJ9&gt;3óz8²*©ãÜ¡+âÃG®¡;å^pY\nÁ|þ7Ü9¿\"®ICmYXqÁÊ½$f,g¶3yÞ!ZéûfÇñÑTU¶ Û\r[M8Ì¨põ)ËåÂxïÑ¬LÂÜlßÜ\nwI×+nX`£åep;·ûÃVÿ À]b").split('.');print r[r.index(input())+1]


The encoding is Latin-1 (aka ISO-8859-1). That first line is important because Python 2 doesn't like non-ASCII characters in files by default (and using Python 3 wouldn't help, because it expects UTF-8 by default, along with loads of other issues).

Takes quoted input (single and double both work), with the initial input as the empty string ('').

The strategy I used was constructing a list of period-separated names (starting with an empty string) and compressing it with zlib (which gave better results over both shoco and lzma). This program decompresses that list, splits it on periods, and outputs the element in the list that follows the input.

I didn't use any real strategy with choosing the next 13 actors, other than picking living British actors (sorry, Alan Rickman, you would've been great) that I liked who haven't played a Doctor yet. The rationale was, the character distribution in the first 13 names is fairly random (as far as names go), so it's unlikely (in my opinion) that 13 more names could be picked that would make a significant difference in compressed size.

For reference, the list I used (with an extra linebreak to separate the new actors):

William Hartnell
Patrick Troughton
Jon Pertwee
Tom Baker
Peter Davison
Colin Baker
Sylvester McCoy
Paul McGann
John Hurt
Christopher Eccleston
David Tennant
Matt Smith
Peter Capaldi

Emma Watson
Patrick Stewart
Ian McKellen
Benedict Cumberbatch
Tom Hiddleston
Sean Connery
Hugh Laurie
Daniel Radcliffe
Rupert Grint
Gary Oldman
Simon Pegg
Keira Knightley
Liam Neeson