Is there a proper way to position manually welded hats?

There are various reasons why the default ROBLOX hat system is untenable for my project, so I've resorted to welding them manually to the desired anatomical parts of my Character (I use a synthetic Head for animation purposes).

I inserted all the Hats that I wanted into a new place using InsertContent, and I'm welding them to the desired Head like-so:

--"hat" is a cloned hat from the original Hat Instances
--"character" is, of course, a reference to the character Model
handle = hat.Handle
hatWeld = Instance.new("Weld", handle)
hatWeld.Part0 = handle
hatWeld.Part1 = head
hatWeld.C1 = CFrame.new(hat.AttachmentPos) --<<--
handle.Parent = character
hat:destroy()

The problem is, that setting the C1 offset of the weld to hat.AttachmentPos doesn't always work - it positions some hats correctly and others not at all correctly (I've tried variants of C0/C1 and +/-attachmentPos, too).

How are we supposed to position Hats correctly using their default properties?