I agree with Slamfire's suggestion to replace the operating rod spring which ArchAngelCD also mentions. While a weak spring would have to be very weak or broken as not to chamber a round meeting specifications into a chamber meeting specifications it never hurts to replace the springs if you aren't sure what you have to begin with.
Now if I assume that the chamber is absolutely clean my best guess is you have a resizing problem with your brass. I doubt going from RCBS dies to Lee dies is the solution. I am also going to leave the discussion of Large Base versus Small Base dies alone. I will say this much on the subject. My niche or one of my niches when I owned a gun shop was The M1 Garand. On average there were 25 of those rifles for sale at any given time in various condition. That went on for years and every rifle I built up was test fired with no less than 48 rounds through it. Needless to say I loaded plenty of 30-06 ammunition for the M1 Garand during those years. I never had a problem loading for the Garand with standard dies, that includes 30-06 for my own Garands as well as .308 for my M1A or my AR10. All this simply means that I personally never had a need for small base sizing dies. So while there may not necessarily be a need for them it sure can't hurt to have them and use them. Just my take on the dies issue.
Next, we can take a look at 30-06 case and chamber specifications.
Here is what I did. I dug into my piles of once fired GI brass in 30-06 and pulled a LC NM case that was fired in a M1 Garand. I cleaned the case and took some measurements on it. The next image shows the case dropped into a M1 Garand barrel. Call it a "plunk test" less the bullet.
The right image shows the brass before sizing and the right image after sizing. Only minor finger force was used for the left image. I am sure using a hammer I could have seated the case deeper and likely using a really big hammer I could have chambered the case. After resizing the case simply dropped into the chamber. The barrel used was an old take off but was shooting OK and did not have excessive headspace.
So let's take a good look at this once fired case and get some dimensions. I have a few tools to measure case dimensions but keep wandering back to my RCBS Precision Micrometer sets. I have checked them against precision chamber headspace gauges and found them to be quite accurate to within the .001" they are graduated in anyway.
RCBS 30-06 Precision Micrometer Set.
Loading the case into the gauge.
Measuring the actual case headspace dimension using the gauge.
We can see at this point that without resizing this case is not going into any 30-06 chamber that meets specifications. Short of using the big hammer I mentioned earlier this case is not going into any M1 Garand chamber known to mankind. The chamber would need to be beyond the realm of Go, No Go, Field and into the realm of "Dearly beloved we are gathered here today...".
Let's look at the cartridge overall length before we re-size it just to get a reference number.
Interesting enough is the actual case overall length is within specification. Take a look at the numbers. Remember those numbers as we will see them again.
Now lets look at the cartridge base dimension just about 0.200" up from the base.
My apologies on this image but believe me we are measuring 0.200" up from the base. Could have done that one better I guess. Anyway, just like the cartridge overall length the base is well within allowable tolerance. Now we will re-size the case using a RCBS FL (Full Length) resizing die.
RCBS FL Resizing Die Set.
Again, as I mentioned before, I could just as easily use a set of Lee dies. The end result would be the same. Also, at this point, knowing the case base diameter I see no reason to use a small base die as the case base is well within specification. The resizing die is set up according to manufacturer procedure and the case is lubricated with case lube.
So let's see what we have using our case gauge.
We now have a case that meets headspace specifications for a 30-06 case. Matter of fact we come up about 0.001" short. Not bad at all and likely why the case freely dropped into the chamber in the second image posted. Knowing the cartridge headspace numbers let's see what has become of the cartridge overall length?
Cartridge overall length following FL resizing.
Interesting as the cartridge overall length did not decrease but actually increased. It not only increased by 0.006" but increased to about 0.001" above specification. This would allow for a nice case trim to exact specification if we so desire.
With everything said and pictured my best guess is that the cases in question are not being properly sized or possibly the cartridge OAL including the bullet on a loaded round is excessive. I would take a sized case and chamber it. Does the bolt go all the way home? Does the rifle go into battery? If you pull the trigger does the rifle go click?
Not pictured and before I forget the final base dimension up 0.200" from the base was 0.4645" or let's call it 0.465" remaining unchanged which would be expected. Since the base was less than 0.4698" to begin with I do not see where a small base die would have mattered in this case.
Just My Take....
Ron