old thread but worth comment- I have a sonic tester and have checked many blocks, various makes, for myself and a few local shops.
the 340 Mopar will go .060" but the resulting cylinder wall thickness may be as low as .095" on the nonthrust walls, between the middle cylinders.
you should really strive to leave .130" remaining all around, for good ring seal.
having said that, the particular 340 I checked, only had .124" on one wall, and .130" on another wall, 2 separate cylinders, even at standard bore. after being bored .030" it had .109" and .115"
I would not bore a 340 .060" unless absolutely necessary, like a rare oem block numbers matched to a car, etc. sure it will run, but my experience with blocks bored to under .100" wall remaining, is they don't make as much power, due to cylinder wall flex. If you can find one at standard bore that would clean up with a hone at standard bore, or only .010" over with .010" pistons, use that.
by the late 1960's all the American car companies went to thinner wall castings compared to the 1950's
V8's, to save on material and lighten up the cars. there are a few exceptions like Oldsmobile, there are 455's that will take .125" over. by contrast, the old 1950's blocks had very thick bores in general, and many of those can go .150" and even .180" I had a 1949 Rocket Olds 303 V8 that I had poked .150" and it could have gone another .030" before hitting .130" wall remaining. the thinnest cylinder walls in that block were over .220" and many were over .250"
the only way to know, is check them with a sonic tester. the 340 I checked, had core shift towards front of block on both banks, making the forward walls thinner than the rear facing walls in every cylinder. there would be .220" on one wall, and .125" on the opposite wall on the same cylinder (non-thrust walls) and all 8 holes were shifted like that. in a perfect world with no core shift, that bore would be centered and have about .170" on both sides, then it would take an .080" overbore and still have .130" wall remaining.
core shift is the final arbiter here, I'd suggest pay the extra $75 or $100 and have it sonic tested first. make sure the operator knows how to calibrate the sonic tester. a part of the block has to be measured with dial calipers or a micrometer, then checked with the sonic tester, then the tester calibrated to that value. otherwise the readings will be invalid.