I shoot off the same style sticks very much.
I don't have the smith skills that some have on this thread but I will add one simple tip that worked on my handi and maybe others can comment as to why it worked so well.
I added a piece of foam weather strip (pressure/harmonics bed) to the forearm tip.
A) Using two "O rings" as listed above on the lug. I took common foam weather strip 3/8" wide x 3/16" thick x roll found at home stores with the sticky back. After coating the inside of my forearm with tung oil to seal out moisture which BTW needed no sanding,no high spots etc. Along with the two O rings, about 1/2" back from the end of the forearm I add the foam strip. Cut approx. 3/4" long and placed across the barrel channel to just slightly come up the sides about 1/3. This combined with my action screw being tight plus 1/4 turn and then marked (torque amount??) gave upward pressure and acts as a vibration isolator. The amount of upward pressure was not measured and is likely less than some recommend but it is definately there to a degree with the two O rings in place. I added two O rings because of the foam strip height. The foam strip compresses to maybe 1/16"-1/8" depending on torque.
I have tested my POI many times and it is spot on (home property rifle range), this was my biggest concern as a varmint hunter. For my gun (Handi 22 Hornet), my 5 shot groups at 100 yards shrunk with the above and proper torque. From my varmint al type shooting sticks, I can flip the sling over the stick and shoot with the gun resting in the Y near the end of the forearm which as you noted is much more comfortable and accurate, with out a poi change. I have tested it.
This foam strip pressure bed is a common sporter rifle accuracy trick to try. I have tried this on other sporter bolt action rifles and did not get the same results as with the Handi. In some previous cases it brought groups in some but, poi changed with humidity etc. Not the case with the Handi , the Handi responded and held POI remarkably well. The Handi was also much more seletive to action screw torque than any other rifle I have owned. IMO- it all adds up to why this simple 15 minute accurizing trick worked well on the Handi. I beieve this to be "harmonics and vibrations"issues mostly with the slight upward pressure being second, the two O rings along with the foam tip pad give it a simple two contact point vibration/float/isolation for lack of better terms. This is simple to add or remove and weather proof. IMO- it is simular to Fred's foam bed concept listed above in mechanics.
A simple tip to try,
Bill / W.S.