The answers to the six sets of clues are: 

1. SEA 
SHEA 
SHAVE 
HEAVES 
HEAVIES

2. O
GO 
HOG 
GOTH 
GHOST

3. F 
IF 
FIX 
FINX 
INFIX

4. TAG 
GNAT 
AGENT 
GANNET 
GENTIAN

5. RAD
DARN 
NADER 
GARDEN 
GRENADE

6. N 
IN 
GIN 
NIGH 
THING

In each set each word is an anagram of the previous word plus one letter.
The letters added in each set are, in order, HVEI GHTS IXNI NENI NEGE
IGHT. (Note that the letters in the first word in each set don't count,
only the added ones.) This spells out HV EIGHT SIX NINE NINE G EIGHT, or
HV8699G8, which is a library call number. The flavortext suggests looking
in the Widener stacks. There are a few books in Widener with this call
number, but only one of them - _Hanging in Judgment_ - was written by
someone named Harry Potter. This book will have a note under the front
cover telling anyone who finds it that the answer to the puzzle is OAR.