I just did clean reinstallations of all my IKM stuff which I had abandoned on my older systems because of constant re-authorization issues.
That seems to have become less of an issue, so I re-installed ST 2.5, 3, and 4 after having taken advantage of the recent VI Group Buy. Everything was better but still weird at times. The IK Product Manager appears to be the way to go, especially if you want to maintain your library/content files off of the C:/System drive (by the time I got done adding to ST3 it was at 151GB and with added content from the recent group buy ST4 is clocking in at 581GB). The product Manager can be set to take care of the authorizations and relocations better than using the separate zip/rar files.

C:\Program Files\IK Multimedia\ (This was listed in the maunual as the default for a SampleTank 3 installation)

C:\Program Files\VSTPlugins\IK Multimedia (This is another place your files could end up)

Most of my newly installed stuff also ended up here with files but not the library content (don't forget, I relocated my libraries)---the directories were way too small for that:
C:\Users\(Your Users Name)\Documents\IK Multimedia
(This would be the exact same location if you just clicked on yout "Documents" icon)
C:\Users\lespa\Documents\IK Multimedia

This looks like where the content originally went (before I relocated it with the Product Manager):
C:\Users\Public\Documents\IK Multimedia

NOW, the contents of my [SampleTank 3] library/content folder looks like this:
S: \ IK Multimedia \ SampleTank 3

Name Size Type
Arpeggios 0 File folder
Effects Presets 0 File folder
Exported Patterns 0 File folder
Imported Instruments0 File folder
Imported Samples 0 File folder
Instruments 0 File folder
Library Info 0 File folder
Multis 0 File folder
Patterns 0 File folder
Samples 0 File folder
Sets 0 File folder
.DS_Store 6,148 DS_STORE File
MultisCache.db 32KB Data Base File
PatternsCache.db 268KB Data Base File
SetsCache.db 166 Data Base File

As far as I can recall, the contents of the [Instruments] and [Samples] should contain these first 20-21 folders:
S: \ IK Multimedia \ SampleTank 3 \ Instruments (the [Samples] folder should be similar)

Name Size Type
01__Acoustic Drums 0 File folder
02__Electronic Drums 0 File folder
03__Bass 0 File folder
04__Synth Bass 0 File folder
05__Acoustic Guitar 0 File folder
06__Electric Guitar 0 File folder
07__Piano 0 File folder
08__Electric Piano 0 File folder
09__Organ 0 File folder
10__Chromatic 0 File folder
11__Strings 0 File folder
12__Brass 0 File folder
13__Woodwinds 0 File folder
14__Synth Lead 0 File folder
15__Synth Pad 0 File folder
16__Voices 0 File folder
17__Ethnic 0 File folder
18__Percussion 0 File folder
19__Sound FX 0 File folder
20__Loops 0 File folder
21__Synth FX 0 File folder

I hope this helps you locate/re-establish ST3 for yourself. Paying the $10 and using the Program Manager would be the simplest route. IKM actually gave me the 180 days when I complained about the fact that it's virtually, if not actually, impossible to purchase the 180 days directly from the website. The website links are circular dead-ends and do not allow you to actually find a place on the website to make the purchase. Apparently, the 180 days have to be purchased via the CustomShop app or not at all. After IK provided me with links that went nowhere, and before anyone realized that it is CustomShop or nothing, they activated 180 days for me as a courtesy for a week of emails.

I will say that, so far, all of the IK stuff is performing well and the authorizations have been maintaining themselves much better than I ever remembered.

Good Luck,
Paj
8^)