James left Cloak with DD - Why?

[Hagrid]

No: HPFGUIDX 155457

Christmas in PS/SS, Harry got James' invisibility cloak with this 
note -

"Your father left this in my possession before he died. It is time 
it was returned to you. Use it well.
A Very Merry Christmas to you."

So why did James leave it with DD? Not for DD to use ...

"I don't need a cloak to become invisible," DD 

Now that we know more about the lead up to that Halloween when James 
and Lily died, does this message seem odd to anyone?

James, Lily and Harry are in hiding, yet leave an invisibility cloak 
with the head of OOTP. James would have found it handy to move 
around outside, but DD must have convinced him someone else needed 
it more than him.

Any ideas? Someone escaping (RAB), or someone spying?