You know, it makes a better joke if the people pouring water on the bale are setting themselves up for it to flame up again tomorrow, if it wouldn’t do that without the water…
So, to put this in the perspective of us city folk…kind of like a mattress in the street, no? They have to be torn apart right down to the springs. The fun part is finding the window of the apartment it was thrown from, so we could drag hose up there and put out the bedroom fire that ensued. :eek: 
Pitch it into space!
Or… surround it with sheets of metal and let it burn, baby.
Or… rope it up and drag it by truck into a body of water.
Peace.
Googling for “hay bale fire” and find instructions to soak the exterior and use a probe to penetrate the bale and introduce water to any hot spots.
I recall having read that fires in bales of cotton could only be put out by kerosene which would penetrate the bale surrounding the fire and exclued the oxygen. Of course care had to be exercised to prevent the outside becoming one big torch! I suspect that “wet water” would do the job today.
Does the VFD get called when this happens? This might work, but would probably be too big of a hassle for just one person to try.
Get a nitrogen cylinder (gas should be common as I’m pretty sure that it’s used as a blanketing agent in silos.) Regulate the output, connect a hose and attach the hose to a length of 1/2" pipe with the other end cut at an angle. Stab the pipe into the center of the bale, turn up the gas and let it suffocate the fire. Once the mess has cooled enough, wrap the bale with plastic to help keep the gas inside. Without oxygen, I would think that the bale would cool down fairly quickly. This would take some experimentation and you would probably still want to fork the bale apart as soon as possible to make sure the fire’s out, but it should cool the bale down in a short time.
If that doesn’t work, then try liquid nitrogen. Hey, if it worked for Red Adair, it should work for you.
Yes, the VFD would typically be called, but whether or not they have had training on any form of specialized fire suppression is a tossup.
This article touches on the use of carbon dioxide or nitrogen for silage extinguishment, but notes that results are not conclusive.
While the introduction of nitrogen via a homemade piercing nozzle would displace oxygen, the cooling effect of the gas would be negligible. Divided combustibles such as hay are capable of storing heat for quite a while, which is why they are prone to rekindling.
Perhaps LN2 would work, but we don’t carry it on fire trucks, and the only time I’ve seen it on a farm is to freeze/store semen. 
Only if you sang: “Roll, roll, roll’n ze hay” ala Teri Garr in Young Frankenstein as you did it.