***Not a teacher***
"Hunt something out" - carries a sense of looking for something amongst other things (literal or figurative).
"Hunt something down" - carries more of a sense of persevering in ones pursuit of something - a determined finding something often despite opposition or initial lack of success.
"Hunt something up" is not an expression I have ever heard used (in BrE). However, I did find it on dictionary.com with the following definition:
"World English Dictionary
hunt up — vb 1. ( tr ) to search for, esp successfully: I couldn't hunt up a copy of it anywhere 2. ( intr ) (of a bell) to be rung progressively earlier during a set of changes
Collins English Dictionary - Complete & Unabridged 10th Edition
2009 © William Collins Sons & Co. Ltd. 1979, 1986 © HarperCollins
Publishers 1998, 2000, 2003, 2005, 2006, 2007, 2009 "
So in answer to your question, it does indeed appear that 'hunt out' and 'hunt up' have very similar meanings.
- For Teachers