Fresh 340 rebuild issue

A ring upside down, especially the 2nd ring, could do that. Or the oil ring spring. But so could a lot of other things as pointed out.And is this oil or water? Some questions:

Does this get worse if you drive it than at idle?

If you coast to a stop, then start out again, is there a large cloud of smoke? That is a classic sign of an oil leak into the intake system.

Is the smoke pure white like snow, or does it have a bluish tinge?

What ring type/brand did you use? Who honed the bores (and to what grit if you know)?

Have you pulled the plugs and checked there color. Post pix of both sets if any look oily or blacl.
I'm leaning towards one second ring being upside down on the left side. I'm questioning myself when I installed the rings on the first few pistons.

Haven't driven it yet, just fired it up last week to break it in and then re-fired to charge the A/C. A third fire showed more blue smoke with added white smoke out of the left side pipe. Again right side clear.

Seems OK until it gets warm and then to be less at idle, then increases as rpm rises but never stops smoking. Again, engine run time has been limited to only 1.5 hours including break-in; not driven at all, still on blocks.

Engine shop bored it to 40 over; I used KB pistons with 4000AM8-080 rings, not sure of grit they used.

Just letting it idle doesn't clear up smoke. Haven't pulled any plugs yet to read them.
Next step in the investigation.

I just don't want to pull the motor if I don't have to but in the same point don't want to ruin the rings or cylinders.

Break-in showed no smoke after 15 minutes, ran real good for 30 minutes.