I have found that on some tenors the G very easily "overblows" to the next harmonic which is a high D. This effect can be created by mechanical problem in which the neck octave pad opens slightly on the high G or can be created by the player himself.
You are correct that there should be a space between the loop from the neck key and the post extending up from the saxophone. However, it is best to do this additional test to make sure the octave mechanism is working correctly.
Checking for a mechanical cause
-Fingering G, press the thumb key hard several times watching the neck octave key.
-If the neck octave key bounces or moves at all, increase the gap between the loop and the post by placing your thumb between the loop and the neck tenon and carefully push down on the neck octave key.
-If the neck octave key opens slightly when you push the thumb key very hard, then a thicker or firmer cork at the area that stops the travel of the thumb key is usually required as well.
-If you go too far and the neck octave doesn't open on high A, place a tongue depressor between the neck pad and the octave pip and then carefully push the loop back toward the neck tenon.
-If the body and neck octave open and close opposite of one another quickly and smoothly when you finger high G to high A and back, then the octave mechanism is working properly.
Checking for a player related cause
-Check the pitch while playing the neck and mouthpiece apart from the sax. The pitch should be close to a Concert E. If the pitch is higher than this the input pitch into the sax is too high and can force unwanted overtones.
-Play a high G and move the back of the tongue up to the "EE" position and blow faster colder air trying to force the high D overtone deliberately.
-When you can produce the D overtone with regularity, practice coming in on high G itself with the tongue in more of an "AH" shape and a slower, warmer airstream.
-When you can control which note starts when you finger high G, then you've wrestled control of that note from the saxophone back to the player where it belongs.