From Maisie Ward's Return to Chesterton (1952), p.63:

A charming story is told by a man who saw Gilbert starting to chase his hat in the street. Plunging through the traffic, the passer-by rescued it to his own imminent peril, and was not pleased when Gilbert received it with the remark that his wife had just bought him a new one and would be sorry to see it again.
"Then why on earth did you run after it?"
"It's an old friend. I am fond of it and I wanted to be with it at the end."