|Conditions obviously change from year to year (and in days) but you can operate on some quite reasonable generalisations. Barron Saddle is usually nice through December then gets too much exposed scree for fun. The upper Murchison is a lovely place but the climbing involves a lot of hiking and is probably a much better area for skiing.
Up the Tasman you can always reach Aylmer, Hochstetter, the rockbuttress on 9144 and the snow walk on Abel (the names of these peaks vary depending on which map you have and to whom you listen). With a minimal amount of zigzagging you climb Green (a fantastic climb) and Walter. The Anna Glacier on Elie de Beaumont is usually cut off in early to mid December, although with a bit of cunning (jumps, abseils) parties get down it a bit later. From mid-December onwards to climb Elie you generally need to traverse Walter twice. Annan and Darwin hardly ever get climbed and with good reason.
You might want to consider going to Centennial on the Franz neve: lovely hut, great setting, rarely busy, limited objective hazard and good access to quite a lot of peaks. This area has several nice, small snow peaks(Aurora, Halcolmb, Aigrette), some quite fun rock scrambling or harder climbing (Rudolf, Spencer, Jervois) -although the divide itself does tend to be quite feral in this section - and a couple of bigger climbs (Conway, Minarets). This is a good area to develop on your experience up the Tasman with a bunch of grade 1-2 routes and the odd harder thing. Conditions and access are usually good well into the season, although Aigrette starts to melt out to crummy rock and it can take some work getting through the schrunds on the Minarets.
Much the same could be said for Pioneer on the Fox neve, although everything is on a bigger scale, except the hut which is smaller and busier. It provides a good further progression from the Franz. A great early season iceclimbing region, it has lots of great snow and rock routes from grade 1-4 on a bunch of bigger peaks that are usually fine with reliable access through February. A great spot to consolidate your skills on a third trip into the hills, or earlier if you are feeling confident.
Aspiring is a great peak to go in and climb but you wouldn't want to spend a whole trip there to consolidate your judgement and skills as there are not that many convenient peaks and the hut is small and constantly overcrowded. From mid-January onwards you would generally expect that the snow lead onto the ramp route might be cut, the northern access onto the NW ridge will have avalanched, access should still be fine via the Quarterdeck (maybe a bit fiddly) or Bevan Col and that conditions in the couloir on the SW ridge could be starting to vary with the storm cycles (it sometimes melts right out to rock, then ices back up etc).
Have a fun trip.