Though .001" is often quoted as the recommended oversize dimension from many sources I actually use it as a minimum. I've run cast bullets .003" over groove diameter on a regular basis without issues. I use the same .311" sizer for bullets for several rifles that range from .308" to .311" so in the tighter bores they are .003" oversized. I've also run some .004" oversized (a .461" cast in a barrel that slugs .457") and didn't have any problems.
My bullets aren't extra hard or anything though. I'm using water quenched pure WW's so probably in the 12-15 bhn range. Many commercial cast bullets are 25+ bhn so they could respond differently.
The only time I've gotten the bullets so large they wont fit in the chamber anymore was when I tried some .314" in a .30-30. I didn't slug the bore but it was a brand new rifle so I would assume somewhere around .3075" to .3085". Assuming it was .308" they were around .006" oversized and too large for the chamber without thinning the neck on the brass. They were dummy rounds and I loaded them specifically to see if they would fit just to learn the limits.
I would buy some .379" for a .376" groove barrel, load a dummy round and see if it chambers. If it does I would try a few out at starting loads, ideally over a chrony to check velocity versus book velocity, and go from there. Check for leading, make sure velocities aren't abnormally high (a sigh of high pressure), and work up the load in the normal manner.