When Martin gets back from the dog park, he's explaining to Daphne about the knitted outfit for Eddie and his own matching hat. You can see the hat sticking out of his pants pocket, but then when they pan back to him pulling it out, he's pulling it out of the coat he had earlier thrown on the couch.
Frasier is shown being against hugging, even being severely awkward about it when trying to hug Niles "for the first time" when Niles is upset about what his mother wrote about him. This is a ridiculous plot for this episode, when he's been shown for the last 4 seasons being a very big touchy feely guy with everyone, including with Niles.