Vintage smiths don’t necessarily shoot better than modern ones do, but people prefer them over modern guns. My 10-6 in 38spl is as accurate as any other gun I own, despite its fixed sight and 4” barrel. It’s also ugly as sin but has an action that is absolutely amazing in both single or double action.
The suggestion of 32 long is also a well respected cartridge but it was always a bit behind the 38sw and by the time 38spl came around it got left in the dust. The 32swl guns are not necessarily expensive, but the good ones aren’t cheap. One with an adjustable sight might run well past a grand, even in other makes. My 30-1 is a J frame gutter sight 3” gun and is amazing when I do my part. I hold the gun and when I hear a pop something has a hole in it, very close to dead center. It’s one of those guns I don’t know when I’m shooting, it just happens at the right time kinda situations.
Now, on to others… as in other brands. There are a few choices, most of which you can rule out as a target gun just by thinking about the purpose behind the company as a whole. Charter for example makes a serviceable pocket gun, but they focused on small and light rather than durable or accurate. Fine for the product that they sell, but not what you want here. NEF/H&R revolvers were made to serve as a cheap alternative to the big names in the wheelgun world. Again they are plenty serviceable, but not specifically intended to be accurate. The only budget gun I could in good conscience recommend is Taurus from the 80s and 90s. The newer ones seem rougher inside and they started having QC issues in the 2000s. Worth noting that most Taurus products are very close copies of the big names visually, and they were even owned by the same company as S&W for a while. The guns either work or they don’t, and the ones that work shoot well.
There was mention of a Ruger single action, which is also a very good platform. The guns are generally made to not only function but be accurate as well, especially the adjustable sight guns. There’s no shame in that single action game, but to be superbly accurate you would need the right version of the gun, not one more tailored to cowboy games.
Long and short, there’s a bazillion old k38s out there with price tags that start with a 3 or 4. Those will work. The ones with adjustable sights will cost more but they are preferred. Colt guns are just about as good but demand a premium price. Not worth the extra in my book. Buy a k38. Shoot wadcutters like suggested. Then buy a j-frame 32 and shoot wadcutters.