I'm heading to Mt Cook region in early Jan. Depends how you gauge easy-moderate, but check out the Ball Pass trek, Mueller Hut/Sealy Range, Sefton Bivy. All involve some steep bits though, the later two being the least committing/easier. Otherwise you can just hike up the Hooker or Tasman glaciers for a more gentle outing.
There is also rock climbing near Mt Cook Village at Sebastapol Bluffs, which I think is mostly bolted and mid graded so minimal rack required.
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