I love the concept of finding hats and using their abilities to return them to the owners. I can see an interesting dilemma of not wanting to return certain hats because they're more useful.

I have a soft spot for platformers, so the concept sounds interesting to me. Based on the document, you have a fairly scalable core mechanic that can result in many interesting level designs. Depending on your intentions, these levels could potentially possess multiple solutions.

My only concern is that the puzzle aspect alongside the range of abilities could result in golden hammer syndrome. For example, a specific hat or a particular combination of hats that have abilities that are so useful that they can be used to effortlessly bypass the level without the others - the high jump hat stands out in particular as the additional jump height could render natural height-based level barriers unintentionally passable with creative platforming. 

However, your schedule is to build all of the abilities first, and then build the levels afterwards. Knowing what abilities are available will make it easier to design, test and readjust the levels. In addition, the document implies that each hat will fulfil a specific niche with little overlap, which will keep each hat relevant for their specific role instead of being replaced by a hat that is more versatile. These two things would render the aforementioned concern moot.

I look forward to seeing what the first set of hats will be.