I just spent 12 months flying to LA and Chicago on the spur of the moment for a health crisis with my mom. While every airline is different, I can tell you without hesitation that the people at Alaska are the kindest and most compassionate I've encountered. They waived change fees, moved flights out by a day when chemo didn't go well and rebanked miles when I had to move travel.
But even with all of that, I never asked them for a low fare for travel the next day or two. Not sure if they'd do anything for you. But I CAN tell you that I became an expert at buying a ticket in the evening and flying out the next morning and that on Alaska, American and United, I was able to buy a cheap seat less than 24 hours before flight time.
It must be related to trying to fill seats. My sister was able to buy a Chicago to LAX for $115 once. I flew between Seattle and LAX for 5000 miles and/or 7500 miles. Click around on their websites and see what you can find for tomorrow.