Both requirements are optional for finishing the original game
A Hat in Time is getting ready to release a major DLC: Seal the Deal, this week.
The first major half, Death Wish (challenge room-esque stuff), is accessible basically with an end game file with 27 time pieces. Anyone who’s finished the main campaign should have it unlocked. The core bit, Arctic Cruise (the new story hub) is a different story. Not only will you need to grab 35 time pieces (more than needed to clear the final boss) you’ll also need the Time Stop Cap: an optional hat that requires 32 yarn to craft.
What I’m saying is, if you want to dive into the core portion of the DLC day one, you’ll want to boot up your file and see where you stand. As a reminder Seal the Deal will be free for the first 24 hours after its September 13 launch. It’ll also include co-op and a 240p filter/costume.