I brought home my 'new' Ruger No.3 in .22 Hornet yesterday, and finished scoping and bore-sighting it this morning.
The rifle came with a set of Weaver bases mounted, which are basically useless since they aren't contoured and sized to match the No. 3's barrel taper. The rear base is OK and I suppose one could mount a red dot on it, but the front base is too low and runs seriously downhill -- even if you changed the contour, it would require a higher ring or a thick shim to work with the rear base.
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Earlier I had tried Ruger's factory bases and rings on my other No.3 in .45-70, but they exhibited their own set of frustrations. The front base required shimming which isn't such a big deal, but the two bases didn't line up very well with the bore, veering noticeably to the right of center. When I tried zeroing a Leupold 1-4x there wasn't enough windage adjustment to get on target.
For scoping the .22 Hornet I decided on a three-pronged approach. I bought a second set of Ruger bases, just in case the first set of bases were the problem rather than the barrel's mounting holes. As backup, I bought a set of Beuhler bases and rings for the No.3 off eBay. And after a lengthy websearch, I also found and bought a Picatinny rail base from EGW that appeared to be compatible with the No. 3:
https://www.egwguns.com/hd-ruger-no-1-varminter-picatinny...
The No.3 Hornet was on layaway for a month, so I did my initial experimentation on the 45-70. It appeared that the Beuhler mounts would work, but the rear sight dovetail interfered with the front base. I'd need to remove it and install a contoured slot blank to preserve aesthetics, so that approach went onto the back burner.
The EGW picrail proved to be a bit too long at both ends to install as-is, but that was an easy fix with a hacksaw and file. I ended up removing two slots from either end to achieve the desired length, and added a little spraypaint to disguise my handiwork. The rail fitted and installed nice and straight on the .45-70's barrel, but had a visible downward cant that I corrected by reusing the front shim I'd made for my first try with Ruger bases. Viola, the shimmed rail lined up just fine, and I was able to successfully mount and zero my Leupold 1-4X. This scope features enough eye relief to safely shoot with hard-kicking loads, should I choose to abuse myself in that manner.
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Knowing that I had at least one working alternative, with the Hornet rifle I began with my newly-purchased set of Ruger bases. Using a pair of rings I had on hand from a No. 1 rifle, I was able to zero my Leupold 2-7x on this rifle. I had to shim the front base once again to get the elevation right, but no big deal.
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The Hornet has better wood than my 45-70, but otherwise the two could be twins. Note that I stuck a Marbles slot blank into the rear sight dovetail -- the folding notch sight was a very loose fit.