The flyer about his itinerary contains the words, "After several days..." And that suggests to me that he planned spending a few days exploring Gardiner Basin, before he headed down Gardiner Creek to Mist Falls. Gardiner Basin is the kind of place where you explore nooks and crannies - it's not the kind of place people want to just rush/march through.
...And where exactly in that big basin he would have been drawn, depends A LOT on what his skills and interests are. That is where his family, friends and hiking partners could help drawing up a profile. From that profile, some of us who've been in the area could venture guesses about where he might have tried to go, the places that most closely meet his interests and his skill level.
Piling onto Mav's questions:
How fast he likes to move - particularly, cross country - could tell us a lot about how many corners of Gardiner Basin he would have had time to explore. A slow or unacclimatized hiker, or someone who just likes to stop and smell the roses, might have only 1 layover day in Gardiner Basin if attempting that route in 7 days. A very fast and skilled hiker could make it into the basin in less than 1.5 days, layover up to 3-4 days and still make it out in his 7-day window. With 3-4 days at his disposal, he could have gotten pretty far up to the top of Gardiner Basin or plausibly even could have left the basin for a time (Sixty Lakes Col; Mt. Clarence King; etc.).
If he is into peak-bagging,
a) Is he only comfortable to Class 3? (and/or, he has climbed Class 4/5 but isn't carrying a rope, and wouldn't dare do any Class 4/5 without a rope?). There is a route from Gardiner Pass to Charlotte Dome, that is characterized as a Class 3. As someone who myself am attracted to fun Class 3 climbs, especially if I can leave my big pack somewhere and just take a daypack, I'd be personally really drawn to climbing such a famous dome. Particularly if I was already planning to go over Gardiner Pass anyway and I was "right there." Based on Cameron's comments about the weather, if Greg was drawn to Charlotte Dome, maybe he hunkered down in his tent for the Tuesday storm, and then tried for Charlotte Dome on Wednesday?
The other draw for a strictly Classes 1-3 peak bagger, is Mount Cotter, which is quite lovely from the middle/upper parts of Gardiner Basin. Class 2/3.
b) Is he into Class 4-5 stuff, and had rope and safety equipment with him? Maybe he tried to climb Mount Gardiner. Or even Mount Clarence King. (King Col is already worth looking into anyway, as Maverick pointed out in an earlier thread. It's the alternative way to get back down to Woods Creek.)
Is he into exploring different corners of lake basins? Is there evidence he knew about the lesser known passes in the basin? Gardiner Pass East - which is located on the northwest shoulder of Mt. Gardiner - would have been a logical route to get him from Gardiner Pass proper directly up to the uppermost basin, where Lake 3477 is. If I had 2+ days available to me in the basin, and was planning the loop he was (eventually descending out Gardiner Creek or even King Col), that is EXACTLY what I would have done. It just creates a logical "loop". However, if he had limited time to do his research, it's possible he may not have even learned about Gardiner Pass East.
He also could have tried to go over a col just to the NE of Gardiner Pass proper, which would have dropped him into one of the side basins, the one where Lake 3223 is. That latter col: I don't even know if it's passable, but Cameron here was just standing on it recently, so maybe he can comment. On the topo map the top of that col looks pretty steep, maybe impassable.
I was just in (upper) Gardiner Basin two weeks ago, August 3-5. My timing didn't overlap with Greg's so I'm sorry I can't be of any direct help. But my impressions of that landscape are still very fresh, so wanted to share my thoughts about what features in this landscape could be the biggest draw for an explorer or peak-bagger.