John, thanks. It's a great tip. Gotta try it sometime. I am very interested in the art of luring hats; it's not something I can find much information about, but seems to have been a vital part of hatmaking.Thanks Stefan!
I just rubbed the oil into the straw of the brim with my fingers. The crown wasn't stiff and dry, so I left it alone. The difference in the pliability was almost immediate. The straw quickly absorbed the small amount I first applied (about 1/4 teaspoon) so I applied about the same amount several hours later, paying attention to getting the woven edge.
The unsettling creaking sensation it gave when bending down the brim is gone, as is the cardboard-like feel and "thonk" as the brim snapped into place. It now seems to have all it's vitality restored.