This is a first for me: I wanted to extend the inspector in a new way and I was able to work out how to do that myself from reading the code :). So I am posting not a question but a tip!
If you want to add a Search button to one specific tabbed view in the inspector (as opposed to above on the object as a whole) then you can do it like this:
composite list "etc"
addAction: someObject gtInspectorActionSpot;
and then you will get a little Search button inside this inspector view that opens an appropriately-sized Spotter on someObject.
I was surprised not to find a 'GTSpotter openOn: someObject' convenience method. Should there be one? There is 'on:' but that leaves you to work out how to open the spotter window with an appropriate size.