68 - depends on what you expect. I've free-floated many barrels, and some improved, and some did not. As far as to 'how much', the dollar bill trick is what I've used.
Just for thought: I free-floated a barrel on a Remington 721 (.30-06) and accuracy improved from 1 1/2" groups at 100 yards to 1 inch with the exact same loads. Did the same for a pre-64 Model 70 Winchester (.30-06) and it made absolutely no difference. In fact, I found that I had better accuracy with the barrel tension screw tightend than I had with the barrel free floated.
I'm not the 'Great Guru Of Rifles' by any measure. What I've found from shooting over the years is the importance of CONSISTANT barrel to stock contact, or lack of contact. Inconsistancey of action to stock, or barrel to stock, contact destroys accuracy.
One of the main reasons a barrel is free floated is to eliminate the variation in the zero with changes in humidity. Those humidity changes cause the stock to swell and distort. At some point the stock touches the barrel hard in one place and as the stock changes shape with the humidity the point of impact is moved as that touch point changes. This is seen as having to change the zero between uses of the gun in a year around time frame. Once you free float the barrel you eliminate that part of the accuracy equation.
But there is more to it that just floating the barrel. Accuracy depends on the stiffness of the barrel, the fit of the stock to the receiver (must not change with recoil or weather or the way you hold the gun). So bedding the stock to the receiver or using a composite stock with a metal bedding block may be the next step.
When the gun is fired the barrel goes through a whipping motion. When you tailor the loads you get the impulse to the barrel to move the barrel so that the bullet leaves the barrel when the barrel is at the extreme of its motion and stopped to go back the other way. That is a function of the velocity and the impulse from the firing. These quite points in the barrel travel can occur at several different velocities of the bullet. You may find loads that are more accurate at several different load levels.
For those who want to know more I recommend the book by Harold R. Vaughn "Rifle Accuracy Facts". This is not a "how-to" book but a study and chronicle of experiments in increasing the accuracy of a rifle. The author has the experience, not in gunsmithing, but in engineering and science to reveal what is really important and what is not and includes some details of design never seen before.
LD is right on. Flaot it out and go test it. Take a couple of strips of plastic cut form a milk jug to the range with you. After youve group fired it floated, loosen the stock and place the strips under the barrel between the stock and barrel at the fore-end tip. then clamp your stock screws back down and group test it again. Some barrels like a bit of upward pressure. In fact I find most actually do.
When you consider the size of the impulse from firing and the few threads that hold the barrel to the receiver and the whipping action of the barrel, it makes sense that supporting a 20+ inch tube of steel might be necessary since the barrel is only held to that receiver by a few threads. (In a screw joint only the last three or so threads actually do all the torquing).
Adding the "pressure point" to the barrel can change the natural frequency of the barrel or at a minimum dampen the oscillations and maybe change the whipping frequency. Once you do that you reintroduce the stock back into the equation, with all its susceptibility to humidity changes. But I am no expert on all of this. The guy that wrote the referenced book became one after doing all the experiments and analyzing the results. I need to re-read the book to go beyond what I have already posted. I may just do that and report back here, if I get the time.
Adding pressure to the fore-end in a sufficient maner does reintroduce the stock, but the pressure negates the stocks warping effect. 03A3s are a prime example. Theres really not a way to successfully float the stock and handguards, so you pressure bed the barrel at the end with at least 15 pounds of upward force. Which essentially amounts to a strip of felt tape beneath the barrel at the fore-end tip and securely clamped with the stock band. So long as the action is solidly bedded so it cannot move around under recoil force it will shoot.
One of my favorite pressure bedding materials is Rigid RTV silicone. Its a time consuming application because you have to smear it on and let it dry in thin layers, then test fit, then smear on another layer and let it dry, and test fit again, up until you have a snug fit with solid compression when you tighten the stock down. Its a vibration dampening material and unaffected by barrel heat.
I have even thought about getting a synthetic stock for this rifle for 2 reasons. I don't want to risk damaging the wood by careless work or hunting abuse. also I would like to play a little with different bedding concepts
While composite stocks with bedding blocks seem like the answer (I have not gone that way myself, yet) a friend bought one (maker unknown to me) and he had to go to glass bedding the interface between the metal bedding block and the receiver. So as with anything else nothings perfect when made by man.
I always likened the rifle barrel to a bell that was "rung" by the firing of the cartridge. Finding an accurate load, like LD pointed out, is done by taking advantage of the natural frequency of the barrel oscillation. Lots of right on information here regarding the rationale for free floating a rifle barrel. As JLA pointed out there are many rifles with either forearms physically attached or upward pressure exerted, and they can be tremendously accurate platforms. Experimentation is the key. Generally though; I find a properly free floated barrel out-performs a non floated barrel if it is accompanied by a good glass bedding job and proper tension on the action screws.
If you want to experiment with bedding (pillars ,glass and such) Boyds is probably the route to go.
If you wan't to try a stock with an aluminum bedding block Stocky's has several iterations in wood or synthetic. the Bell and Carlson's are good synthetic stocks with the bedding blocks. Use Stocky's stock finder in the upper left hand corner to find what they have available.