Another vote for the M27, M28/30 or M39 Mosin-Nagant from Finland.
If you like to tinker, you can make most M91/30s shoot about as well. You must remember that most 91/30s we see here are refurbished with unfitted parts, and they need work.
Two-stage triggers may be had, as well as receiver shimming kits (a type of bedding), improved sights, lots of stuff.
A vote, too, for one I've not seen mentioned, the M88 and its derivatives. Mine cleaned up nicely after setting in the corner of a gunshop for a couple years. Got it for $37.50 OTD. The gunshop owner didn't know what it was and wasn't interested in messing with C&R.
The bore was rusty and the lands are slightly rounded. It was made in 1892 in Amberg, Germany.
When I first shot it, it was 1.2MOA. I've not measured groups since.
The whole platform was built around precision. It has a sleeved barrel and very fine sights. If you have good eyesight, the rifle shoots well. If not, you'll have trouble.
A downside is that you must handload. I do have the "S" conversion, which means mine was modified to take the then-new 150grn spitzer round. The differences between the M88 7.92x57mm and the "JS" (or "IS" if you prefer) 7.92x57mm are a threefold: 1. Bullet type and weight, down from 220grn or so, 2. velocity, mostly due to reduced bullet weight and 3. bullet diameter, up to 0.323" from the original 0.318" of the M88 round.
You cannot shoot modern 8x57mm in it. Pressures are too high. You'd have to reload as I do, or buy M88 ammo from Europe, where it's recognized as different rounds and not lumped together.
The M88 is an excellent rifle. It's just a matter of whether you're willing to go to the trouble of manufacturing or finding the proper ammo.
Regards,
Josh