When JJ grabbed his coat and scarf getting ready to confront Carl abt why he dumped his mom, JJ left WITH NO HAT.
When the Thelma and Michael were talking to the police on the phone regarding "JJ missing" they described JJ to a tee, even with the denim hat. But when JJ and Carl walked in to the apt. JJ didn't have HIS DENIM HAT. He had no hat. He only had the ice bag on his head. Shortly before JJ's look-a-like walked in, all of a sudden JJ had his denim hat on while also having the ice pack on his head.